L’Hôpital’s rule practice question 22

When $x$ is huge, how large can $\sin x$ be next to $x$? Judge each choice by what the quotient itself does, whichever method you use.

The form is $\tfrac{\infty}{\infty}$, but the rule gives $\dfrac{1 + \cos x}{1}$, which keeps swinging between 0 and 2. When the new limit does not exist, the rule says nothing about the original one.

Divide the top and the bottom by $x$ instead: $1 + \dfrac{\sin x}{x}$. The numerator $\sin x$ stays between $-1$ and $1$ while $x$ grows.