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'.)