can I use a sequence to determine function limit

Question

Suppose I have to find the limit of function at $x_0 $, can I use sequence instead ?

( I wrote the divirtev of f but the question not only in that case)

\begin{gather*} \lim _{x\rightarrow x_{0}}\frac{f( x) -f( x_{0})}{x-x_{0}} =\lim\limits _{n\rightarrow \infty }\frac{f( x_{n}) -f( x_{0})}{x_{n} -x_{0}}\\ \\ \text{where : }\;\lim _{n\rightarrow \infty } x_{n} =x_{0} \end{gather*}

If yes what I need to check before changing the limit and when in what cases that is possible?

I think that it is ok but I can't find a way to explain that .

Answer 1

Given a function $h$, a point $x_0$, and a number $l$ asserting that $\lim_{x\to x_0}h(x)=l$ is equivalent to asserting that, whenever a sequence $(x_n)_{n\in\mathbb N}$ converges to $x_0$, the sequence $\bigl(f(x_n)\bigr)_{n\in\mathbb N}$ converges to $l$.

Therefore, the following assertions are equivalente:

• $f'(x_0)=l$;
• for every sequence $(x_n)_{n\in\mathbb N}$ converges to $x_0$,$$\lim_{n\to\infty}\frac{f(x_n)-f(x_0)}{x_n-x_0}=l.$$