What's the difference between the two？

Question

Question

Which of the following two can define the derivative $f'(a)$:

1)$$\lim_{n \to \infty}n \left[f\left(a+\frac{1}{n}\right)-f(a)\right],n \in \mathbb{Z}.$$

2)$$\lim_{x \to \infty}x\left[f\left(a+\frac{1}{x}\right)-f(a)\right],x \in \mathbb{R}.$$

Answer 1

Let $f(x)= \begin{cases} 0 & \text{If $x$ is rational.} \\ 1 & \text{If $x$ is irrational} \end{cases}$

Then $\displaystyle \lim_{\substack{n \to \infty} \\ n \in \mathbb{Z}}n \left[f\left(0+\frac{1}{n}\right)-f(0)\right] = 0$

and

$\displaystyle \lim_{\substack{n \to \infty} \\ n \in \mathbb R \setminus \mathbb{Q}}n \left[f\left(0+\frac{1}{n}\right)-f(0)\right] = \infty$.

Hence

$\displaystyle \lim_{\substack{n \to \infty} \\ n \in \mathbb R}n \left[f\left(0+\frac{1}{n}\right)-f(0)\right]$ does not exist.