Introduction Introduction

A limit is a statement of the form:
As

x
approaches
a
,
f(x)
approaches
c
.

Some example limits for the graph below
(f(x)=\frac{\sin(x)}{x})
are: As
x
approaches
0
,
f(x)
approaches
1
.

As
x
approaches
-\infty
,
f(x)
approaches
0
.

As
x
approaches
\infty
,
f(x)
approaches
0
.


\frac{\sin(0)}{0}
is undefined.
However, the limit still exists because the following holds: As
x
approaches
0
,
\sin(x)/x
approaches
1
.
Limits only exist if:
The limit from the left and the limit from the right are the same.
The limit is finite (not
-\infty
or
\infty
).
The function is not oscillating. (Will be shown later)


Limits can exist even if:
The function itself does not exist.
The limit is at
-\infty
or
\infty
.

Examples

Below are graphs which do not have limits when

x
approaches
0
:

The limit does not exist at
x=0
because
f(x)
approaches
\infty
.

Note that for the statement:

As x approaches

a
,
f(x)
approaches
c

a
can be
\infty
, but
c
cannot be.
The limit does not exist at
x=0
because
f(x)
has a different limit from the left and the right.
The limit does not exist at
x=0
because the function is oscillating.
The limit does not exist at
x=0
because the function does not have a limit from the left.

Practice

Finding a limit at an

x
-value
(x=a)
means filling in the blank:
As
x
approaches
a
, f(x) approaches ____.


To do this, we do not use the value of
f(a)
. Instead, we use the values of
f(x)
as
x
gets extremely close to
a
, and then we try to guess what
f(a)
is.
For the following, use the first text box to submit
x
-values. Using
f(x)
for those
x
-values, find the limit and submit it in the second text-box. When the limit is guessed correctly, a graph will display.

(If the limit does not exist, type 'undefined'.)

f() =

f(10)=

f() =

f(\infty)=

f() =

f(0)=

f() =

f(0)=

f() =

f(1)=

f() =

f(0)=