Originally Posted by

**dannyboycurtis** I am having trouble verifying the following problem:

If the functions $\displaystyle f$ and $\displaystyle g$ are defined on $\displaystyle (a,\infty)$ with $\displaystyle a\in \mathbb{R}$, and where $\displaystyle \lim_{x\to \infty} f(x) = L$ and $\displaystyle \lim_{x\to \infty} g(x) = +\infty$, prove that $\displaystyle \lim_{x\to \infty} (f \circ g)(x) = L$

Here is what I have tried to do

Suppose $\displaystyle (x_{n})$ is a sequence defined on $\displaystyle (a,\infty)$ which diverges to $\displaystyle \infty$.

At this point I need to show that the sequence $\displaystyle (g(x_{n}))$ diverges to $\displaystyle \infty$, thereby confirming that the sequence $\displaystyle (f(g(x_{n})))$ converges to L as $\displaystyle n\to \infty$.

How can I show that the sequence $\displaystyle (g(x_{n}))$ diverges?