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Problem 54 Hard Difficulty

(a) Evaluate $ \lim_{x \to ^x} x \sin \frac {1}{x}. $
(b) Evaluate $ \lim_{x \to 0} x \sin \frac{1}{x}. $
(c) Illustrate parts (a) and (b) by graphing $ y = x \sin (1/x). $

Answer

a) 1
b) $\lim _{x \rightarrow 0} x \sin \frac{1}{x}=0$
c) SEE GRAPHS

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Video Transcript

It's Clara suing you. Read here. So we have the limit as X approaches. Infinity, X sign one over X. This is equal to the limit as X approaches Infinity. We're sign of one over. Max over one over X. He substitutes one over X. You for one over X. So we get the limit. As you approach is, zero were signed. You over you. We know that the limit estate approaches zero for signed data over. Data is equal to one. So this has to be equal to one for part B. We're gonna use something called the sandwich here. Um, we want to find out the limit. Starting with X approaches. Syrup positive for a sign one over X. You know, that sign won over. Axe is between negative one and one included, and we're gonna multiply X to get negative. X sign one over X Positive X. So we get the limit as ex coaches. Positive zero. The X sign one over Max is equal to zero Now we want to find it from the negative side. So we get negative X next time. Sign one over x them positive X. You can be right. This as X I'm sign one over X. It's less than or equal to negative X. So using the sandwich, dear, um, this is equal to zero. So we have shown that limits are both equal to zero. So the limit as X approaches zero X sign went over acts is equal 20 for part C. We're gonna draw a graph and goes like this where the limit as X approaches infinity. One graph will look something like this. And as X approaches zero, the grass grows down more.