impossible to
say
The Squeeze theorem allows us to compute the limit of a difficult function by ”squeezing” it between two easy functions.
Consider the function 
Is this function continuous at ?
We must show that . First, let’s assume that and
small. Since
by multiplying these inequalities by , we obtain which can be written as
and, therefore, as
Now, let’s assume that and small. Since by multiplying these inequalities by , we obtain which can be written as
Recall that Therefore for all small values of x
Since we apply the Squeeze Theorem and obtain that
. Hence is continuous.
Here we see how the informal definition of continuity being that you can “draw it” without “lifting your pencil” differs from the formal definition.
Compute:
To compute this limit, use the Squeeze Theorem. First note that we only
need to examine and for the present time, we’ll assume that is positive. Consider
the diagrams below:
 |  |  |
From our diagrams above we see that and computing these areas we find Multiplying through by , and recalling that we obtain Dividing through by and taking the reciprocals (reversing the inequalities), we find Note, and , so these inequalities hold for all . Additionally, we know and so we conclude by the Squeeze Theorem, .
When solving a problem with the Squeeze Theorem, one must write a sort of mathematical poem. You have to tell your friendly reader exactly which functions you are using to “squeeze-out” your limit.
Compute:
Let’s graph this function to see what’s going on: 
Remark: Since it follows that
Hence we have that when and we see and so by the Squeeze theorem, In a similar
fashion, when , and so and again by the Squeeze Theorem . Hence we see that
This remark is very important, since the function has two factors: