## anonymous one year ago Find the limit: lim as x approaches 0 of cos(1/x)

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1. anonymous

DNE

2. SolomonZelman

$$\large\color{slate}{\displaystyle\lim_{x \rightarrow ~0}\cos\left(\frac{1}{x}\right)}$$

3. SolomonZelman

That, does not exist

4. anonymous

@pgpilot326 can you explain?

5. anonymous

@solomonzelman why?

6. Zale101

If you sub in x=0, you'll get a 0 in the denominator which is undefined.

7. SolomonZelman

$$\large\color{slate}{\displaystyle\lim_{x \rightarrow ~0}\cos\left(\frac{1}{x}\right)}=\cos\left(\displaystyle \lim_{x \rightarrow ~0}\frac{1}{x}\right)$$

8. anonymous

as x approaches 0, 1/x approaches infinity. cos will cycles through all of it's values and not settle on a single value (which it would need to do in order for the limit to exist)

9. SolomonZelman

yes, that is equivalent of $$\large\color{slate}{\displaystyle\lim_{x \rightarrow ~0}\cos\left(\frac{1}{x}\right)}=\cos\left(\displaystyle \lim_{x \rightarrow ~0}\frac{1}{x}\right)=\cos\left(\displaystyle \lim_{x \rightarrow ~\infty }x\right)$$

10. SolomonZelman

So it will alternate between 1 and -1

11. SolomonZelman

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