Thread: [SOLVED] Limit of a composition of functions

1. [SOLVED] Limit of a composition of functions

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?

2. 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?
Didn't you post this exact question and get an answer here?

3. So I did, thank you, I seem to have forgotten. Sorry for the repetition.