Question

Find the limit:
lim as x approaches 0 of cos(1/x)

Answer 1

DNE

Answer 2

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

Answer 3

That, does not exist

Answer 4

@pgpilot326 can you explain?

Answer 5

@solomonzelman why?

Answer 6

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

Answer 7

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

Answer 8

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)

Answer 9

yes, that is equivalent of

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

Answer 10

So it will alternate between 1 and -1