The Squeeze theorem allows us to compute the limit of a difficult function by “squeezing” it between two easy functions.
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.
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.
This remark is very important, since the function has two factors: