Squeeze that Function

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The limit laws in calculus only apply when the limits exist. This means other types of limit theorems or results must be used to work with the more difficult type of limit problems.

The following problem taken from the fourth edition of Calculus Concepts and Contexts by James Stewart is an example of a hard limit that can not be solved using the limit laws. It is problem 32 in section 2.3.

The first thing to observe about this particular limit is that the sine of pi divided by x does not have a limit as x approaches zero from the right, but we do know that the sin function is always between -1 and 1. Now if we exponentiate both sides of the inequality you see below, we squeeze the square root of x times e raised to the sin of pi over x between two functions that have a limit of zero.

So we have pinched, squeezed or sandwiched the given function between two simpler functions that have known limit of zero which means the given function must have a limit of zero!

Finally, remember to try the Squeeze Theorem when faced with a challenging limit question and I hope the picture of the Maple Version 16 graph of  the function being squeezed is a helpful visual.

Squeeze Theorem Problem from Stewart fourth Edition
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