+<%= @person[:description] %>
diff --git a/app/views/home/about.html.erb b/app/views/home/about.html.erb new file mode 100644 index 0000000..1da72a0 --- /dev/null +++ b/app/views/home/about.html.erb @@ -0,0 +1,6 @@ +<%=raw html_logo %> is an experiment in mathematics.
+Need help with a math course at Ohio State? Looking for lecture videos? Looking for practice exams? This is the place. <%= @course %>
-<%= link_to 'Login', user_omniauth_authorize_path(:google_oauth2), +
<%= link_to 'Login', user_omniauth_authorize_path(:coursera), { :class => "btn btn-primary btn-large" }%>
<%=raw html_logo %> was produced by a bunch of people. Here are some of them.
+<%=raw html_logo %> could not have been produced without the generous support of the mathematics department at The Ohio State University.
+ +Uses Twitter’s bootstrap and Khan Academy’s exercise framework. Built on Rails.
+Given that f(INITIALX) = INITIALY, and f(FINALX) = FINALY,
- approximate f'(INITIALX).
-
Suppose that f(INITIALX) = INITIALY, and f(FINALX) = FINALY.
+
Approximate f'(INITIALX).
+
SLOPE
Remember that f'(INITIALX) is the slope of the best linear approximation
- to f near x = INITIALX
Remember that f'(INITIALX) is the slope of the best linear approximation to f near x = INITIALX.
We only have enough information to get the slope of the secant line passing through the points
(INITIALX,INITIALY) and (FINALX,FINALY),
but this should be close to the slope of the tangent line to f at x = INITIALX
because INITIALX and FINALX are close.
The slope of a line is the ratio of the change in ordinates (\Delta y) to the change in abscissas (\Delta x)
The change in ordinates is \Delta y = FINALY - INITIALY = DELTAY
The change in abscissas is \Delta x = FINALX - INITIALX = DELTAX
The slope of a line is the ratio of the change in ordinates (\Delta y) to the change in abscissas (\Delta x).
The change in ordinates is \Delta y = FINALY - INITIALY = DELTAY.
The change in abscissas is \Delta x = FINALX - INITIALX = DELTAX.
The slope of the secant line from (INITIALX,INITIALY) to (FINALX,FINALY)
- is \frac{\Delta y}{\Delta x} = \frac{DELTAY}{DELTAX} = SLOPE
Since the slope of the tangent line is close to the slope of the secant line, f'(INITIALX) \approx SLOPE
+ is \frac{\Delta y}{\Delta x} = \frac{DELTAY}{DELTAX} = SLOPE.
Since the slope of the tangent line is close to the slope of the secant line, f'(INITIALX) \approx SLOPE.
- Consider the function f(x) graphed below. Find
- the value of each of the indicated limits with an error of at most \frac{1}{2}.
-
-
Consider the function f(x) graphed below.
+
Find the value of each of the indicated limits with an error of at most \frac{1}{2}.
+
+ +
\displaystyle\lim_{x \to LIMITX^{-}}f(x):
ANSWER
+
\displaystyle\lim_{x \to LIMITX^{+}}f(x):
ANSWER
+
\displaystyle\lim_{x \to LIMITX}f(x):
ANSWER
-
- Consider the function f(x) graphed below. Find
- the value of each of the indicated limits with an error of at most \frac{1}{2}.
-
-
Consider the function f(x) graphed below.
+
Find the value of each of the indicated limits with an error of at most \frac{1}{2}.
+
\displaystyle\lim_{x \to LIMITX^{-}}f(x):
ANSWER
\displaystyle\lim_{x \to LIMITX^{+}}f(x):
ANSWER+JUMP
\displaystyle\lim_{x \to LIMITX}f(x):
DNE
@@ -188,9 +179,9 @@
- Consider a function f(x). While
+
+ Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
values of x, provided that x
- \ne LIMITX. Find{}^*
+ \ne LIMITX.
+
+
+ Find{}^*
the value of each of the indicated limits
by evaluating f(x) at various values,
or state that the limit does not exist (DNE).
-
f()= ?
\displaystyle\lim_{x \to LIMITX^{-}}f(x):
@@ -72,22 +72,22 @@
Try values of x close to LIMITX but not equal to LIMITX
Try values of x close to LIMITX but not equal to LIMITX.
f(LIMITX-.1) = POLY.evalOf(LIMITX-.1)
f(LIMITX-.001) = POLY.evalOf(LIMITX-.001)
f(LIMITX-.000001) = POLY.evalOf(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the left.
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER.
Now repeat the process for values of x which are greater than LIMITX.
f(LIMITX+.1) = POLY.evalOf(LIMITX+.1)
f(LIMITX+.001) = POLY.evalOf(LIMITX+.001)
f(LIMITX+.000001) = POLY.evalOf(LIMITX+.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the right.
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER.
Since the limits from both the right and the left exist and are equal,
\displaystyle\lim_{x \to LIMITX}f(x) = ANSWER
- Consider a function f(x). While
+
Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
values of x, provided that x
- \ne LIMITX. Find{}^*
+ \ne LIMITX.
+
Find{}^*
the value of each of the indicated limits
by evaluating f(x) at various values,
or state that the limit does not exist (DNE).
-
f()= ?Try values of x close to LIMITX but not equal to LIMITX
Try values of x close to LIMITX but not equal to LIMITX.
f(LIMITX-.1) = POLY.evalOf(LIMITX-.1)
f(LIMITX-.001) = POLY.evalOf(LIMITX-.001)
f(LIMITX-.000001) = POLY.evalOf(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the left.
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER.
Now repeat the process for values of x which are greater than LIMITX.
f(LIMITX+.1) = JUMP+POLY.evalOf(LIMITX+.1)
f(LIMITX+.001) = JUMP+POLY.evalOf(LIMITX+.001)
f(LIMITX+.000001) = JUMP+POLY.evalOf(LIMITX+.000001)
It looks like f(x) is approaching JUMP+ANSWER as x approaches LIMITX
from the right.
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = JUMP+ANSWER
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = JUMP+ANSWER.
The limits from both the right and the left exist, but they are not equal, so
\displaystyle\lim_{x \to LIMITX}f(x) does not exist (DNE).
- Consider a function f(x). While
+
+ Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
values of x, provided that x
- \ne LIMITX. Find{}^*
+ \ne LIMITX.
+
+ Find{}^*
the value of each of the indicated limits
by evaluating f(x) at various values,
or state that the limit does not exist (DNE).
-
f()= ?Try values of x close to LIMITX but not equal to LIMITX
Try values of x close to LIMITX but not equal to LIMITX.
f(LIMITX-.1) = POLY.evalOf(LIMITX-.1)
f(LIMITX-.001) = POLY.evalOf(LIMITX-.001)
f(LIMITX-.000001) = POLY.evalOf(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the left.
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER.
Now repeat the process for values of x which are greater than LIMITX.
f(LIMITX+.1) = ANSWER+FACTOR*Math.sin(1/(.1))
f(LIMITX+.01) = ANSWER+FACTOR*Math.sin(1/(.01))
f(LIMITX+.000001) = ANSWER+FACTOR*Math.sin(1/(.000001))
It appears that f(x) is not settling down to any one value as x approaches LIMITX from the right.
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) does not exist (DNE)
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) does not exist (DNE).
Even though the limit exists as x approaches LIMITX from the left,
the limit from the right does not exist, so the two sided limit
\displaystyle\lim_{x \to LIMITX}f(x) also does not exist (DNE).
@@ -263,20 +261,20 @@
- Consider a function f(x). While
+
+ Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
values of x, provided that x
- \ne LIMITX. Find{}^*
+ \ne LIMITX.
+
+ Find{}^*
the value of each of the indicated limits
by evaluating f(x) at various values,
or state that the limit does not exist (DNE).
-
f()= ?Try values of x close to LIMITX but not equal to LIMITX
Try values of x close to LIMITX but not equal to LIMITX.
f(LIMITX-.1) = ANSWER+FACTOR*Math.sin(-1/(0.1))
f(LIMITX-.01) = ANSWER+FACTOR*Math.sin(-1/(.01))
f(LIMITX-.001) = ANSWER+FACTOR*Math.sin(-1/(.001))
@@ -332,36 +330,36 @@
f(LIMITX-.000001) = ANSWER+FACTOR*Math.sin(-1/(.000001))
It appears that f(x) is not settling down to any one value as x approaches LIMITX from the left.
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) does not exist (DNE)
Now repeat the process for values of x which are greater than LIMITX
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) does not exist (DNE).
Now repeat the process for values of x which are greater than LIMITX.
f(LIMITX+.1) = POLY.evalOf(LIMITX+.1)
f(LIMITX+.001) = POLY.evalOf(LIMITX+.001)
f(LIMITX+.000001) = POLY.evalOf(LIMITX+.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the right.
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER.
Even though the limit exists as x approaches LIMITX from the right,
the limit from the left does not exist, so the two sided limit
- \displaystyle\lim_{x \to LIMITX}f(x) also does not exist (DNE).
-
\displaystyle\lim_{x \to LIMITX}f(x) also does not exist (DNE).
- Consider a function f(x). While
+
+ Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
values of x, provided that x
- \ne LIMITX. Find{}^*
+ \ne LIMITX.
+
+
+ Find{}^*
the value of each of the indicated limits
by evaluating f(x) at various values,
or state that the limit does not exist (DNE).
-
f()= ?Try values of x close to LIMITX but not equal to LIMITX
Try values of x close to LIMITX but not equal to LIMITX.
f(LIMITX-.1) = ANSWER+FACTOR*Math.sin(-1/(0.1))
f(LIMITX-.01) = ANSWER+FACTOR*Math.sin(-1/(.01))
f(LIMITX-.001) = ANSWER+FACTOR*Math.sin(-1/(.001))
@@ -414,8 +412,8 @@
f(LIMITX-.000001) = ANSWER+FACTOR*Math.sin(-1/(.000001))
It appears that f(x) is not settling down to any one value as x approaches LIMITX from the left.
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) does not exist (DNE)
Now repeat the process for values of x which are greater than LIMITX
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) does not exist (DNE).
Now repeat the process for values of x which are greater than LIMITX.
f(LIMITX+.1) = ANSWER+FACTOR*Math.sin(1/(.1))
f(LIMITX+.01) = ANSWER+FACTOR*Math.sin(1/(.01))
f(LIMITX+.001) = ANSWER+FACTOR*Math.sin(1/(.001))
@@ -424,7 +422,7 @@
f(LIMITX+.000001) = ANSWER+FACTOR*Math.sin(1/(.000001))
It appears that f(x) is not settling down to any one value as x approaches LIMITX from the right.
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) does not exist (DNE)
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) does not exist (DNE).
Both the limit from the left and the right fail to exist, so the two sided limit
\displaystyle\lim_{x \to LIMITX}f(x) also does not exist (DNE).
- Consider a function f(x). While
+
+ Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
values of x, provided that x
- \ne LIMITX. Find{}^*
+ \ne LIMITX.
+
+ Find{}^*
the value of each of the indicated limits to at least 3 digits of accuracy
by evaluating f(x) at various values,
or state that the limit does not exist (DNE).
-
f()= ?Try values of x close to LIMITX but not equal to LIMITX
Try values of x close to LIMITX but not equal to LIMITX.
f(LIMITX-.1) = f(LIMITX-.1)
f(LIMITX-.001) = f(LIMITX-.001)
f(LIMITX-.000001) = f(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the left.
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER.
Now repeat the process for values of x which are greater than LIMITX.
f(LIMITX+.1) = f(LIMITX+.1)
f(LIMITX+.001) = f(LIMITX+.001)
f(LIMITX+.000001) = f(LIMITX+.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the right.
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER
Hence \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER.
Since the limits from both the right and the left exist and are equal,
- \displaystyle\lim_{x \to LIMITX}f(x) = ANSWER
+ \displaystyle\lim_{x \to LIMITX}f(x) = ANSWER.
- Consider a function f(x). While
+
+ Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
values of x, provided that x
- \ne LIMITX. Find{}^*
+ \ne LIMITX.
+
+
+ Find{}^*
the value of each of the indicated limits to at least 3 digits of accuracy
by evaluating f(x) at various values,
or state that the limit does not exist (DNE).
-
f()= ?Try values of x close to LIMITX but not equal to LIMITX
Try values of x close to LIMITX but not equal to LIMITX.
f(LIMITX-.1) = f(LIMITX-.1)
f(LIMITX-.001) = f(LIMITX-.001)
f(LIMITX-.000001) = f(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the left.
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER.
Now repeat the process for values of x which are greater than LIMITX.
f(LIMITX+.1) = JUMP+f(LIMITX+.1)
f(LIMITX+.001) = JUMP+f(LIMITX+.001)
f(LIMITX+.000001) = JUMP+f(LIMITX+.000001)
It looks like f(x) is approaching JUMP+ANSWER as x approaches LIMITX
from the right.
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = JUMP+ANSWER
Hense \displaystyle\lim_{x \to LIMITX^{+}}f(x) = JUMP+ANSWER.
The limits from both the right and the left exist, but they are not equal, so
\displaystyle\lim_{x \to LIMITX}f(x) does not exist (DNE).
f is a continuous function. While you don't have
+ a formula for f, you can evaluate f for various values below.
+ f(leftBound) \approx roundTo(3,f(leftBound)) and
+ f(rightBound) \approx roundTo(3,f(rightBound)). By the intermediate value theorem, there must be
+ at least one root of f between leftBound and rightBound. We have insured that their is
+ only one zero in that interval.
+
+ Use the bisection method to give 3 consecutively better approximations of the zero. + Each approximation should be the midpoint of the interval you choose. +
+ ++
f()= ?
+ First approximation:
+ ANSWERS[0]
+
+ Second approximation:
+ ANSWERS[1]
+
+ Third approximation:
+ ANSWERS[2]
+
hint
+ +
+ The function f is graphed below. Is f continuous at
+ x=LIMITX? If not, why not?
+
Continuous
+\displaystyle \lim_{x \to LIMITX} f(x) \neq f(LIMITX)\displaystyle \lim_{x \to LIMITX} f(x) does not exist Try values of x close to LIMITX but not equal to LIMITX
f(LIMITX-.1) = f(LIMITX-.1)
f(LIMITX-.001) = f(LIMITX-.001)
f(LIMITX-.000001) = f(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
+ from the left.
+
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
f(LIMITX+.1) = f(LIMITX+.1)
f(LIMITX+.001) = f(LIMITX+.001)
f(LIMITX+.000001) = f(LIMITX+.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
+ from the right.
+
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER
Since the limits from both the right and the left exist and are equal,
+ \displaystyle\lim_{x \to LIMITX}f(x) = ANSWER
+
+ The function f is graphed below. Is f continuous at
+ x=LIMITX? If not, why not?
+
Discontinuous, because \displaystyle \lim_{x \to LIMITX} f(x) \neq f(LIMITX)
\displaystyle \lim_{x \to LIMITX} f(x) \neq f(LIMITX)\displaystyle \lim_{x \to LIMITX} f(x) does not exist Try values of x close to LIMITX but not equal to LIMITX
f(LIMITX-.1) = f(LIMITX-.1)
f(LIMITX-.001) = f(LIMITX-.001)
f(LIMITX-.000001) = f(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
+ from the left.
+
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
f(LIMITX+.1) = f(LIMITX+.1)
f(LIMITX+.001) = f(LIMITX+.001)
f(LIMITX+.000001) = f(LIMITX+.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
+ from the right.
+
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER
Since the limits from both the right and the left exist and are equal,
+ \displaystyle\lim_{x \to LIMITX}f(x) = ANSWER
+
+ The function f is graphed below. Is f continuous at
+ x=LIMITX? If not, why not?
+
Discontinuous, because \displaystyle \lim_{x \to LIMITX} f(x) does not exist
\displaystyle \lim_{x \to LIMITX} f(x) \neq f(LIMITX)\displaystyle \lim_{x \to LIMITX} f(x) does not exist Try values of x close to LIMITX but not equal to LIMITX
f(LIMITX-.1) = f(LIMITX-.1)
f(LIMITX-.001) = f(LIMITX-.001)
f(LIMITX-.000001) = f(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
+ from the left.
+
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
f(LIMITX+.1) = JUMP+f(LIMITX+.1)
f(LIMITX+.001) = JUMP+f(LIMITX+.001)
f(LIMITX+.000001) = JUMP+f(LIMITX+.000001)
It looks like f(x) is approaching JUMP+ANSWER as x approaches LIMITX
+ from the right.
+
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = JUMP+ANSWER
The limits from both the right and the left exist, but they are not equal, so
+ \displaystyle\lim_{x \to LIMITX}f(x) does not exist (DNE).
+
- Consider a function f(x). While
+
+ Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
- values of x. Does f appear to be continuous or discontinuous at
+ values of x.
+
+
+
+ Does f appear to be continuous or discontinuous at
x=LIMITX, and why?
-
f()= ?Continuous
We need to see if \displaystyle \lim_{x \to LIMITX} f(x) = f(LIMITX)
We need to see if \displaystyle \lim_{x \to LIMITX} f(x) = f(LIMITX).
f(LIMITX)= ANSWER
Now we need to see if \displaystyle \lim_{x \to LIMITX} f(x) exists and is equal to f(LIMITX)
Try values of x close to LIMITX but not equal to LIMITX
Now we need to see if \displaystyle \lim_{x \to LIMITX} f(x) exists and is equal to f(LIMITX).
Try values of x close to LIMITX but not equal to LIMITX.
f(LIMITX-.1) = f(LIMITX-.1)
f(LIMITX-.001) = f(LIMITX-.001)
f(LIMITX-.000001) = f(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the left.
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
Hense \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER.
Now repeat the process for values of x which are greater than LIMITX.
f(LIMITX+.1) = f(LIMITX+.1)
f(LIMITX+.001) = f(LIMITX+.001)
f(LIMITX+.000001) = f(LIMITX+.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the right.
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER
Hense \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER.
Since the limits from both the right and the left exist and are equal,
- \displaystyle\lim_{x \to LIMITX}f(x) = ANSWER
+ \displaystyle\lim_{x \to LIMITX}f(x) = ANSWER.
- Consider a function f(x). While
+
+ Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
- values of x. Does f appear to be continuous or discontinuous at
+ values of x.
+
+
+
+ Does f appear to be continuous or discontinuous at
x=LIMITX, and why?
-
f()= ?Discontinuous, because \displaystyle \lim_{x \to LIMITX} f(x) \neq f(LIMITX)
Try values of x close to LIMITX but not equal to LIMITX
Try values of x close to LIMITX but not equal to LIMITX.
f(LIMITX-.1) = f(LIMITX-.1)
f(LIMITX-.001) = f(LIMITX-.001)
f(LIMITX-.000001) = f(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the left.
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
Hense \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER.
Now repeat the process for values of x which are greater than LIMITX.
f(LIMITX+.1) = f(LIMITX+.1)
f(LIMITX+.001) = f(LIMITX+.001)
f(LIMITX+.000001) = f(LIMITX+.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the right.
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER
Hense \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER.
Since the limits from both the right and the left exist and are equal,
- \displaystyle\lim_{x \to LIMITX}f(x) = ANSWER
+ \displaystyle\lim_{x \to LIMITX}f(x) = ANSWER.
- Consider a function f(x). While
+
+ Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
- values of x. Does f appear to be continuous or discontinuous at
+ values of x.
+
+
+ Does f appear to be continuous or discontinuous at
x=LIMITX, and why?
-
f()= ?Discontinuous, because \displaystyle \lim_{x \to LIMITX} f(x) does not exist
Try values of x close to LIMITX but not equal to LIMITX
Try values of x close to LIMITX but not equal to LIMITX.
f(LIMITX-.1) = F1(LIMITX-.1)
f(LIMITX-.001) = F1(LIMITX-.001)
f(LIMITX-.000001) = F1(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the left.
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
Hense \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER.
Now repeat the process for values of x which are greater than LIMITX.
f(LIMITX+.1) = F1(LIMITX+.1)
f(LIMITX+.001) = F1(LIMITX+.001)
f(LIMITX+.000001) = F1(LIMITX+.000001)
It looks like f(x) is approaching JUMP+ANSWER as x approaches LIMITX
from the right.
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = JUMP+ANSWER
Hense \displaystyle\lim_{x \to LIMITX^{+}}f(x) = JUMP+ANSWER.
Even though the limits from the left and right both exist, they are not equal.
- \displaystyle\lim_{x \to LIMITX}f(x) Does Not Exist (DNE)
+ \displaystyle\lim_{x \to LIMITX}f(x) Does Not Exist (DNE).
- Consider a function f(x). While
+
+ Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
- values of x. Does f appear to be continuous or discontinuous at
+ values of x.
+
+
+ Does f appear to be continuous or discontinuous at
x=LIMITX, and why?
-
f()= ?Discontinuous, because \displaystyle \lim_{x \to LIMITX} f(x) does not exist
Try values of x close to LIMITX but not equal to LIMITX
Try values of x close to LIMITX but not equal to LIMITX.
f(LIMITX-.1) = F2(LIMITX-.1)
f(LIMITX-.001) = F2(LIMITX-.001)
f(LIMITX-.000001) = F2(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the left.
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
Hense \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER.
Now repeat the process for values of x which are greater than LIMITX.
f(LIMITX+.1) = F2(LIMITX+.1)
f(LIMITX+.001) = F2(LIMITX+.001)
f(LIMITX+.000001) = F2(LIMITX+.000001)
f(x) does not appear to be approaching any one value as x
approaches LIMITX from the right. (try other values even closer to LIMITX)
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) does not exist. (DNE)
Hense \displaystyle\lim_{x \to LIMITX^{+}}f(x) does not exist (DNE).
If either of the one sided limits do not exist, the limit does not exist.
- \displaystyle\lim_{x \to LIMITX}f(x) Does Not Exist (DNE)
+ \displaystyle\lim_{x \to LIMITX}f(x) Does Not Exist (DNE).
- Consider a function f(x). While
+
+ Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
- values of x. Does f appear to be continuous or discontinuous at
+ values of x.
+
+
+
+ Does f appear to be continuous or discontinuous at
x=LIMITX, and why?
-
f()= ?Discontinuous, because \displaystyle \lim_{x \to LIMITX} f(x) does not exist
Try values of x close to LIMITX but not equal to LIMITX
Try values of x close to LIMITX but not equal to LIMITX.
f(LIMITX-.1) = F3(LIMITX-.1)
f(LIMITX-.001) = F3(LIMITX-.001)
f(LIMITX-.000001) = F3(LIMITX-.000001)
f(x) does not appear to be approaching any one value as x
approaches LIMITX from the left. (try other values even closer to LIMITX)
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) does not exist. (DNE)
Now repeat the process for values of x which are greater than LIMITX
Hense \displaystyle\lim_{x \to LIMITX^{-}}f(x) does not exist (DNE).
Now repeat the process for values of x which are greater than LIMITX.
f(LIMITX+.1) = F3(LIMITX+.1)
f(LIMITX+.001) = F3(LIMITX+.001)
f(LIMITX+.000001) = F3(LIMITX+.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
from the right.
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER
Hense \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER.
If either of the one sided limits do not exist, the limit does not exist.
- \displaystyle\lim_{x \to LIMITX}f(x) Does Not Exist (DNE)
+ \displaystyle\lim_{x \to LIMITX}f(x) Does Not Exist (DNE).
- Consider a function f(x). While
+
+ Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
- values of x. Does f appear to be continuous or discontinuous at
+ values of x.
+
+
+
+ Does f appear to be continuous or discontinuous at
x=LIMITX, and why?
-
f()= ?Discontinuous, because \displaystyle \lim_{x \to LIMITX} f(x) does not exist
Try values of x close to LIMITX but not equal to LIMITX
Try values of x close to LIMITX but not equal to LIMITX.
f(LIMITX-.1) = F3(LIMITX-.1)
f(LIMITX-.001) = F3(LIMITX-.001)
f(LIMITX-.000001) = F3(LIMITX-.000001)
f(x) does not appear to be approaching any one value as x
approaches LIMITX from the left. (try other values even closer to LIMITX)
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) does not exist. (DNE)
Now repeat the process for values of x which are greater than LIMITX
Hense \displaystyle\lim_{x \to LIMITX^{-}}f(x) does not exist (DNE).
Now repeat the process for values of x which are greater than LIMITX.
f(LIMITX+.1) = F2(LIMITX+.1)
f(LIMITX+.001) = F2(LIMITX+.001)
f(LIMITX+.000001) = F2(LIMITX+.000001)
f(x) does not appear to be approaching any one value as x
approaches LIMITX from the right. (try other values even closer to LIMITX)
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) does not exist. (DNE)
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) does not exist (DNE).
If either of the one sided limits do not exist, the limit does not exist.
- \displaystyle\lim_{x \to LIMITX}f(x) Does Not Exist (DNE)
+ \displaystyle\lim_{x \to LIMITX}f(x) Does Not Exist (DNE).
+ Consider a function f(x). While
+ you don't know the formula for f(x)
+ you can nevertheless still evaluate it for given
+ values of x, provided that x
+ \ne LIMITX. Find{}^*
+ the value of each of the indicated limits to at least 3 digits of accuracy
+ by evaluating f(x) at various values,
+ or state that the limit does not exist (DNE).
+
+
f()= ?
+ {}^*You can't really be sure whether you are finding the limit from
+ only a finite amount of data, but the function is "reasonable" enough that your experimental data
+ will not lead you astray.
+
Continuous
+\displaystyle \lim_{x \to LIMITX} f(x) \neq f(LIMITX)\displaystyle \lim_{x \to LIMITX} f(x) does not exist Try values of x close to LIMITX but not equal to LIMITX
f(LIMITX-.1) = f(LIMITX-.1)
f(LIMITX-.001) = f(LIMITX-.001)
f(LIMITX-.000001) = f(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
+ from the left.
+
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
f(LIMITX+.1) = f(LIMITX+.1)
f(LIMITX+.001) = f(LIMITX+.001)
f(LIMITX+.000001) = f(LIMITX+.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
+ from the right.
+
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER
Since the limits from both the right and the left exist and are equal,
+ \displaystyle\lim_{x \to LIMITX}f(x) = ANSWER
+
+ Consider a function f(x). While
+ you don't know the formula for f(x)
+ you can nevertheless still evaluate it for given
+ values of x, provided that x
+ \ne LIMITX. Find{}^*
+ the value of each of the indicated limits to at least 3 digits of accuracy
+ by evaluating f(x) at various values,
+ or state that the limit does not exist (DNE).
+
+
f()= ?
+ {}^*You can't really be sure whether you are finding the limit from
+ only a finite amount of data, but the function is "reasonable" enough that your experimental data
+ will not lead you astray.
+
Discontinuous, because \displaystyle \lim_{x \to LIMITX} f(x) \neq f(LIMITX)
\displaystyle \lim_{x \to LIMITX} f(x) \neq f(LIMITX)\displaystyle \lim_{x \to LIMITX} f(x) does not exist Try values of x close to LIMITX but not equal to LIMITX
f(LIMITX-.1) = f(LIMITX-.1)
f(LIMITX-.001) = f(LIMITX-.001)
f(LIMITX-.000001) = f(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
+ from the left.
+
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
f(LIMITX+.1) = f(LIMITX+.1)
f(LIMITX+.001) = f(LIMITX+.001)
f(LIMITX+.000001) = f(LIMITX+.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
+ from the right.
+
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = ANSWER
Since the limits from both the right and the left exist and are equal,
+ \displaystyle\lim_{x \to LIMITX}f(x) = ANSWER
+
+ Consider a function f(x). While
+ you don't know the formula for f(x)
+ you can nevertheless still evaluate it for given
+ values of x, provided that x
+ \ne LIMITX. Find{}^*
+ the value of each of the indicated limits to at least 3 digits of accuracy
+ by evaluating f(x) at various values,
+ or state that the limit does not exist (DNE).
+
+
f()= ?
+ {}^*You can't really be sure whether you are finding the limit from
+ only a finite amount of data, but the function is "reasonable" enough that your experimental data
+ will not lead you astray.
+
Discontinuous, because \displaystyle \lim_{x \to LIMITX} f(x) does not exist
\displaystyle \lim_{x \to LIMITX} f(x) \neq f(LIMITX)\displaystyle \lim_{x \to LIMITX} f(x) does not exist Try values of x close to LIMITX but not equal to LIMITX
f(LIMITX-.1) = f(LIMITX-.1)
f(LIMITX-.001) = f(LIMITX-.001)
f(LIMITX-.000001) = f(LIMITX-.000001)
It looks like f(x) is approaching ANSWER as x approaches LIMITX
+ from the left.
+
so \displaystyle\lim_{x \to LIMITX^{-}}f(x) = ANSWER
Now repeat the process for values of x which are greater than LIMITX
f(LIMITX+.1) = JUMP+f(LIMITX+.1)
f(LIMITX+.001) = JUMP+f(LIMITX+.001)
f(LIMITX+.000001) = JUMP+f(LIMITX+.000001)
It looks like f(x) is approaching JUMP+ANSWER as x approaches LIMITX
+ from the right.
+
so \displaystyle\lim_{x \to LIMITX^{+}}f(x) = JUMP+ANSWER
The limits from both the right and the left exist, but they are not equal, so
+ \displaystyle\lim_{x \to LIMITX}f(x) does not exist (DNE).
+
- Consider a function f(x). While
+
+ Consider a function f(x). While
you don't know the formula for f(x)
you can nevertheless still evaluate it for given
- values of x. Given that f'(X) is an integer,
+ values of x.
+
+
+ Given that f'(X) is an integer,
approximate f'(X).
- A calculator may be necessary
-
f()= ?ANS
The definition of f'(X) is
- \displaystyle\lim_{h \to 0} \frac{f(X + h)-f(X)}{h}
\displaystyle\lim_{h \to 0} \frac{f(X + h)-f(X)}{h}.
Try values of h which are close to, but not equal to zero.
+ Consider the function
+ hint
+ Consider the function
+ hint
+ Consider the function
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+ hint
+ Consider the function
+ hint
+ Consider the function
+ hint
+ Consider the function
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Consider the function
+
+ hint
+ Find
+
+
+ The expression is of the form
+ To do this, the first step is to find two values
+ The next step is to rewrite the
+ expression as To apply the first step, you need to find
+ The following values can be used: For the second step, rewrite the expression as: or The next step is to factor both terms of the above expression:
+
+ Redistribute the common term to get the answer:
+
+
+
+ A fully completed limit computation is presented below. Carefully check each step of the calculation, and either say which
+ limit law justifies the step, or that a mistake has been made. Note that we use the abbreviation "nbhd" for the word "neighborhood"
+
+
+
+
+
diff --git a/curriculum/khanExercise/exercises/infiniteLimitsGraphical.html b/curriculum/khanExercise/exercises/infiniteLimitsGraphical.html
new file mode 100644
index 0000000..edf8776
--- /dev/null
+++ b/curriculum/khanExercise/exercises/infiniteLimitsGraphical.html
@@ -0,0 +1,250 @@
+
+
+
+
+ f(x) graphed below. Find
+ the value of each of the indicated limits to within \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ \displaystyle\lim_{x \to LIMITX^{-}}f(x):
+ leftSol
+
+ \displaystyle\lim_{x \to LIMITX^{+}}f(x):
+ rightSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ \displaystyle\lim_{x \to LIMITX^{-}}f(x):
+ leftSol
+
+ \displaystyle\lim_{x \to LIMITX^{+}}f(x):
+ g(LIMITX)
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ \displaystyle\lim_{x \to LIMITX^{-}}f(x):
+ g(LIMITX)
+
+ \displaystyle\lim_{x \to LIMITX^{+}}f(x):
+ rightSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits to three decimal places of accuracy.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to LIMITX^{-}}f(x):
+ leftSol
+
+ \displaystyle\lim_{x \to LIMITX^{+}}f(x):
+ rightSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to LIMITX^{-}}f(x):
+ leftSol
+
+ \displaystyle\lim_{x \to LIMITX^{+}}f(x):
+ g(LIMITX)
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to LIMITX^{-}}f(x):
+ g(LIMITX)
+
+ \displaystyle\lim_{x \to LIMITX^{+}}f(x):
+ rightSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits to three decimal places of accuracy.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to LIMITX^{-}}f(x):
+ leftSol
+
+ \displaystyle\lim_{x \to LIMITX^{+}}f(x):
+ rightSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to LIMITX^{-}}f(x):
+ leftSol
+
+ \displaystyle\lim_{x \to LIMITX^{+}}f(x):
+ g(LIMITX)
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to LIMITX^{-}}f(x):
+ g(LIMITX)
+
+ \displaystyle\lim_{x \to LIMITX^{+}}f(x):
+ rightSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits to three decimal places of accuracy.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to LIMITX^{-}}f(x):
+ leftSol
+
+ \displaystyle\lim_{x \to LIMITX^{+}}f(x):
+ rightSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to LIMITX^{-}}f(x):
+ leftSol
+
+ \displaystyle\lim_{x \to LIMITX^{+}}f(x):
+ g(LIMITX)
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to LIMITX^{-}}f(x):
+ g(LIMITX)
+
+ \displaystyle\lim_{x \to LIMITX^{+}}f(x):
+ rightSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ \displaystyle\lim_{x \to +\infty}f(x):
+ p.coefs[4]
+
+ \displaystyle\lim_{x \to -\infty}f(x):
+ C
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ \displaystyle\lim_{x \to +\infty}f(x):
+ posSol
+
+ \displaystyle\lim_{x \to -\infty}f(x):
+ negSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ \displaystyle\lim_{x \to +\infty}f(x):
+ p.coefs[4]
+
+ \displaystyle\lim_{x \to -\infty}f(x):
+ negSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ \displaystyle\lim_{x \to +\infty}f(x):
+ posSol
+
+ \displaystyle\lim_{x \to -\infty}f(x):
+ p.coefs[4]
+ f(x). Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to +\infty}f(x):
+ p.coefs[4]
+
+ \displaystyle\lim_{x \to -\infty}f(x):
+ C
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to +\infty}f(x):
+ posSol
+
+ \displaystyle\lim_{x \to -\infty}f(x):
+ negSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to +\infty}f(x):
+ p.coefs[4]
+
+ \displaystyle\lim_{x \to -\infty}f(x):
+ negSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to +\infty}f(x):
+ posSol
+
+ \displaystyle\lim_{x \to -\infty}f(x):
+ p.coefs[4]
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to +\infty}f(x):
+ p.coefs[4]
+
+ \displaystyle\lim_{x \to -\infty}f(x):
+ C
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to +\infty}f(x):
+ posSol
+
+ \displaystyle\lim_{x \to -\infty}f(x):
+ negSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to +\infty}f(x):
+ p.coefs[4]
+
+ \displaystyle\lim_{x \to -\infty}f(x):
+ negSol
+ f(x) graphed below. Find
+ the value of each of the indicated limits with an error of at most \frac{1}{2}.
+ If one of the limits appears to be \infty or -\infty, just type "infinity"
+ or "-infinity".
+ f()= ?\displaystyle\lim_{x \to +\infty}f(x):
+ posSol
+
+ \displaystyle\lim_{x \to -\infty}f(x):
+ p.coefs[4]
+ \displaystyle\lim_{x\toLIMITX}\frac{format(PROBLEM, "large")}{format(PROBLEM2, "large")}
+ \displaystyle\lim_{x\toLIMITX}f(x):
+ roundTo(2, SOL)
+
+ parseFormat("#{A}x^2+#{B}x+#{C}}", [BLUE, PINK, ORANGE]).
+ You can factor this trinomial by grouping. In this case,
+ \blue{A}=\blue{A}, \pink{B}=\pink{B},
+ \orange{C}=\orange{C}
+ \green{a} and
+ \green{b}, such that:parseFormat("#{ab} = #{A}*#{C}", [GREEN, BLUE, ORANGE])parseFormat("#{a}+#{b} = #{B}", [GREEN, GREEN, PINK])parseFormat("#{A}x^2 + #{a}x + #{b}x + #{C}", [BLUE, GREEN, GREEN, ORANGE]) or
+ parseFormat("(#{A}x^2 + #{a}x) + (#{b}x + #{C})", [BLUE, GREEN, GREEN, ORANGE]), and factor each term, to
+ find a common factor.\green{a} and \green{b} such that:formatGroup(GROUP1, [0,1])parseFormat("#{a}=#{" + a + "}", [GREEN, GREEN])
+ parseFormat("#{b}=#{" + b + "}", [GREEN, GREEN])parseFormat("#{" + A + "}x^2+#{" + a + "}x+#{" + b + "}x+#{" + C + "}}", [BLUE, GREEN, GREEN, ORANGE])parseFormat("(#{" + A + "}x^2+#{" + a + "}x)+(#{" + b + "}x+#{" + C + "})", [BLUE, GREEN, GREEN, ORANGE])format(HINT1, {del1factors:true, evalBasicNumOps:true})
+ format(, simplifyOptions.basic, false, "large")
+ \frac{num.text()}{denom.text()}
+
+
+
+ \displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()}
+
+
+
+
+
+ \hphantom{\displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()=}}
+
+
+
+
+
+
+
+ = \displaystyle\displaystyle\lim_{x \to A}\frac{(lin1.text())(lin2.text())}{(lin1.text())(lin3.text())}
+
+
+
+
+ \hphantom{\displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()=}}
+
+
+
+
+
+
+
+ = \displaystyle\lim_{x \to A}\frac{lin2.text()}{lin3.text()}
+
+
+
+
+
+ \hphantom{\displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()=}}
+
+
+
+
+
+
+
+
+ = \displaystyle\frac{\displaystyle\lim_{x \to A}(lin2.text())}{\displaystyle\lim_{x \to A}(lin3.text())}
+
+
+
+
+
+ \hphantom{\displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()=}}
+
+
+
+
+
+
+
+
+ = \displaystyle\frac{\displaystyle\lim_{x \to A}(Dx)+\displaystyle\lim_{x \to A}(-B*D)}
+ {\displaystyle\lim_{x \to A}(Ex)+\displaystyle\lim_{x \to A}(-C*E)}
+
+
+
+
+
+ \hphantom{\displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()=}}
+
+
+
+
+
+
+
+
+ = \displaystyle\frac{D\displaystyle\lim_{x \to A}(x)+\displaystyle\lim_{x \to A}(-B*D)}
+ {E\displaystyle\lim_{x \to A}(x)+\displaystyle\lim_{x \to A}(-C*E)}
+
+
+
+
+
+ \hphantom{\displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()=}}
+
+
+
+
+
+
+
+
+ = \displaystyle\frac{(D)(A)+\displaystyle\lim_{x \to A}(-B*D)}
+ {(E)(A)+\displaystyle\lim_{x \to A}(-C*E)}
+
+
+
+
+
+ \hphantom{\displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()=}}
+
+
+
+
+
+
+
+
+ = \displaystyle\frac{(D)(A)+-B*D}
+ {(E)(A)+-C*E}
+
+
+
+
+
+ \hphantom{\displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()=}}
+
+
+
+
+
+
+
+
+ = \displaystyle\frac{D*A-B*D}
+ {E*A-C*E}
+
+
+
+
+
+