Can one apply LHopitals' rule to differentiable functions defined over the naturals?

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.



Answers 1


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.

David Reed
David Reed
January 19, 2018 02:23 AM

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