by Saad
Last Updated January 19, 2018 02:20 AM

For e.g, if we have $\lim_{n \rightarrow \infty} \frac {f(n)}{g(n)}$= $\frac 00$, $f:R \rightarrow R$ and $g:R \rightarrow R$ (note that $f$,$g$ are defined on $R$ so the derivative makes sense) so in essence, we're considering sequences.

As mentioned in the comments, the answer is yes. More generally, there is an "equivalence" between limits of functions and limits of sequences. In particular, the following is a theorem:

$$\lim_{x \to a}f(x) = L \text { if and only if for every sequence } \{x_n\} \text{ the following is true:}$$ $$ \text { if } \lim_{n \to \infty}x_n = a \text{ then } \lim_{n \to \infty} f(x_n)= L$$

Here we actually require $x_n \neq a$. This still holds true when $a,L \in \{\pm \infty\}$

EDIT- Just in the interests of being tedious, I should mention that we have to restrict $x_n$ to be in some open interval containing $a$ (or having $a$ as an endpoint if $a = \pm \infty$) on which $f$ is defined.

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