Evaluating limits by subsituting special sequences, justification for that Sometimes I saw people using transformations like
$$
 \lim_{x\to 0} f(x) = \lim_{n\to \infty} f(\frac{1}{n})
$$
or
$$
 \lim_{x \to p} f(x) = \lim_{n \to \infty} f(x + \frac{1}{n}). \quad (*)
$$
I know that
$$
  \lim_{x \to p} f(x) = q \Leftrightarrow \lim_{n\to \infty} f(p_n) = q
$$
for all sequences $p_n \to p$. So how is it justified to use such a special sequence like $p_n = \frac{1}{n}$ for example in (*)?

Take the function $f(x)=\sin\left(\frac{2\pi}{x}\right)$. Now we have:
$$
\lim_{n\to\infty}f(\frac 1n)=\lim_{n\to\infty}\sin\left(2\pi n\right)=0 \\
\lim_{n\to\infty}f(\frac 4{4n+1})=\lim_{n\to\infty}\sin\left(\frac{(4n+1)\pi }{2}\right)=\lim_{n\to\infty}\sin\left(\frac{\pi }{2}\right)=1 
$$
So if the limit of $f$ at $x$ does not exist at that point then not all different sequences converging to $x$ have the same limit.
