Computing Limits
In the previous
section we saw that there is a large class of function that allows us to use
to compute limits. However, there are also many limits for
which this won�t work easily. The purpose of this section is to develop
techniques for dealing with some of these limits that will not allow us to just
use this fact.
Let�s first got back and take a look at one of the first
limits that we looked at and compute its exact value and verify our guess for
the limit.
Example 1
Evaluate the following limit.
Solution
First let�s notice that if we try to plug in
 we
get,
So, we can�t just plug in
 to
evaluate the limit. So, we�re going to have to do something else.
The first thing that we should always do when evaluating
limits is to simplify the function as much as possible. In this case that means
factoring both the numerator and denominator. Doing this gives,
So, upon factoring we saw that we could cancel an
 from
both the numerator and the denominator. Upon doing this we now have a new
rational expression that we can plug
 into
because we lost the division by zero problem. Therefore, the limit is,
Note that this is in fact what we guessed the limit to be.
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