Question

Find the limit of f(x) as x approaches infinity:

f(x)=cos(1/x)/(1+(1/x))
if you can help please explain each step.

Answer 1

i think you get 1 by inspection

Answer 2

meaning
\[\lim_{x\rightarrow \infty}\frac{1}{x}=0\] as since cosine is continuous this means
\[\lim_{x\rightarrow \infty}\cos(\frac{1}{x})=\cos(0)=1\]

Answer 3

likewise
\[\lim_{x\rightarrow \infty}(1+\frac{1}{x})=1+0=1\]so the whole thing is
\[\frac{1}{1}=1\]

Answer 4

think all the steps are there. "by inspection" meant no tricks or l'hopital's rule

Answer 5

so you found the limit of the numerator and the limit of the denominator and made that the answer? you can do that?

Answer 6

yes

Answer 7

you can do it unless you get
\[\frac{0}{0}\] then you have to do more work

Answer 8

right. teacher emphasized sandwich theorem in lecture so i've attempted to find it that way with no luck. do you know how to do it that way?