I have calculus homework question that to be quite frank I don't even know how to begin to solve. Here is my attempt on how to to solve the question; however I'm not sure if I'm tackling this correctly. I'm hoping that someone here can tell me if I'm doing this correctly and if my answer would be correct.
Here is the question.
Find the limit or show that it does not exist:
$$\lim_{x\to \infty}(e^{-x}+2\cos3x)$$
My attempt to the solution is DNE or does not exist simply because cosine oscillates between -1 and 1 and it never approaches any one number to reach a horizontal asymptote. Is this answer correct? Or how should I approach this problem if it's not? Or should it be
$$-\infty$$ since that would be the the value of $$e^{-x}\quad ?$$Please forgive my stupidity if I'm incorrect in all fronts.
Best Answer
Two things you need to notice. The $e^{-x}$ term decays to zero as $x\to\infty$.
The other term oscillates between $2$ and $-2$. So the whole sum oscillates between a little more than $2$ and a little more than $-2$.