As n approaches infinity, what is the limit of (2+sin(n^2))/(42+(1/n))

I'll type a cleaner version in the comments

Question

As n approaches infinity, what is the limit of (2+sin(n^2))/(42+(1/n))

I'll type a cleaner version in the comments

mhchen · Answer

$$\lim_{n \rightarrow \infty}\frac{2+\sin(n^2)}{42+\frac{1}{n}}$$

Vocaloid · Answer

@angle (my calculus is very rusy but iirc the limit of sin(x^2) DNE as x --> infinity b/c the function fluctuates between -1 and 1?)

Angle · Answer

correct

Vocaloid · Answer

so the entire limit would not exist? :S

Angle · Answer

exactly

if any part of a limit Does Not Exist (DNE)
then the DNE trumps all and the entire limit cannot exist

mhchen · Answer

ah.....doesn't exist then. That's my problem.

mhchen · Answer

Wait so if I'm trying to compare 2 functions on which one has a 'higher dominance', like little-o.
Say function f(x) and g(x).

I'd do 
$$\lim_{x \rightarrow \infty}\frac{f(x)}{g(x)}$$

If it becomes 0, g(x) dominates. If it becomes infinity, f(x) dominates, if it's a constant, they have the same theta-notation.

If they don't exist....would they also have the same theta notation?

Angle · Answer

I think that it might be that the function that causes the DNE would dominate

mhchen · Answer

hmm, that would make more sense