How to calculate a limit when x goes to infinity without using l'hospital 's rule.

Question

How to find the limit of $$\lim_{x \to \infty} 3\cos(2x +1)+ [1/(2x +1)^3 ] - 1$$ when $x$ goes to infinity. When I am doing this question I couldn't able to find the limit of cos(2x +1). Should I put a constant for cos(2x +1) or can I calculate the limit without using a constant ?

Answer 1

The limit does not exist.

If we assume the limit exists, then $\lim\limits_{x \to \infty} \cos x$ must exist. However, the limit does not exist, as it moves between $-1$ or $1$.

So the answer is the limit does not exist.