Q:

Why doesn't L'Hospital's rule work for this limit?

Let $a,b,A,B$ be positive real numbers, with $a>b$ and $A>B$. Consider the limit:
$$\lim_{x\to\infty}\frac{ax+b\sin(x)}{Ax+B\sin(x)}$$
By the squeeze theorem, the limit exists and is equal to $\frac{a}{A}$, but applying L'Hospital's rule we find
$$\lim_{x\to\infty}\frac{ax+b\sin(x)}{Ax+B\sin(x)}=\lim_{x\to\infty}\frac{a+b\cos(x)}{A+B\cos(x)}$$
The limit on the right-hand side of the inequality above does not exist. What is going wrong here?

A:

If the limit of derivatives exists, then they're equal. If the limit of derivatives doesn't exist, then you can tell nothing 

