I recently came across this question which was closed as being a duplicate of that question.

But if we read carefully, we see that there is a slight difference.

In "this question", it's supposed that: $\forall n \in \mathbb N \ , \ x_n<y_n$

In "that question", it's supposed that: $\forall n \in \mathbb N \ , \ \alpha_n < 0 < \beta_n$

And so, the answers to the two questions are different.

In "this question", we may have $\lim_{n\rightarrow \infty} \dfrac{f(y_n)-f(x_n)}{y_n-x_n} \neq f'(0)$ if $f'$ is not continuous.

In "that question", $\lim_{n\rightarrow \infty} \dfrac{f(\beta_n)-f(\alpha_n)}{\beta_n-\alpha_n} = f'(0)$ even if $f'$ is not continuous.

1. I'm not fluent in english. I would be grateful to anyone who edits this post to correct it.
2. What should have I done instead of using Meta to expose this problem ?