Limit as x tends towards +infinity for cos(1/x) to be 1

[grizz45]

I have worked out the limit as x tends towards +infinity for cos(1/x) to be 1, as cos (1/infinity) would be cos(0) which is 1. However the answer in the book with the question says that the answer should be (0,3) and (3,+infinty)! Is this a misprint or have i gine drastically wrong?

 

[rasmhop]

I have worked out the limit as x tends towards +infinity for cos(1/x) to be 1, as cos (1/infinity) would be cos(0) which is 1. However the answer in the book with the question says that the answer should be (0,3) and (3,+infinty)! Is this a misprint or have i gine drastically wrong?

Of course
\lim_{x \to \infty} \cos(1/x)
doesn't tend toward (0,3) and (3,infty) (how would that make any sense?). That is probably the answer to a different problem (have you checked the problem number and chapter number is correct?). Your idea is correct except for the fact that 1/infinity makes no sense, but since cos is continuous and defined in 0 you have:
\lim_{x \to \infty} 1/x = 0 \Rightarrow \lim_{x \to \infty} \cos(1/x) = \cos(0) = 1
so your answer is correct.

 

[grizz45]

i have checked the problem number and everything...i think there is a massive misprint as i have done some other questions from the same section and the answers all seem to be wrong as there are of the same format as the answer in the first post!