Theorem stating the connection of limits of sequences and functions.

Question

Following is the theorem I have been given for regarding the connection between limits of sequences and functions, in order to help decide whether a limit for a function does exist or not:

Theorem: Suppose that $f$ is a real function, defined on $(a−R, a)∪(a, a+R)$ for some $R \in R^+$. Then $ \lim_{x\to a} f(x) = \ell$ if and only if $\lim_{n \to \infty}f(a_n) =\ell $, for all sequences $(a_n)$ such that $\lim_{n \to \infty}=a$ and $a_n \neq a$ for all $n \in N$.

My question is whether $a = \infty $ is allowed?

Also, does anyone know of any resources that can help me out with these questions, with examples and advice of how to choose such sequences, that will give the counter example necessary to show that the limit does not exist?

Answer 1

It is true that if $\lim_{x\to\infty} f(x)= l$ then $\lim_{n\to\infty} f(x_n)= l$ for any sequence, $\{x_n\}$ such that $\lim_{n\to\infty} x_n= \infty$.

It is also true that if $\lim_{n\to \infty} f(x_n)= l$ for every sequence $\{x_n\}$ such that $\lim_{n\to\infty} x_n= \infty$ then $\lim_{x\to\infty} f(x)= l$.