Example where $\lim_{x\to 0}f(x^2)$ exists but $\lim_{x\to 0}f(x)$ does not. Can somebody give me an example where $$\lim_{x\to 0}f(x^2)$$ exists but $$\lim_{x\to 0}f(x)$$ does not?

Yes.  Let $f(x) = 1$ for $x<0$ and $f(x) = 0$ for $x \geq 0$.  $f(x^2) = 0$ so the limit exists.  $f(x)$ is has a jumpt discontinuity at zero (the limit from the left is $1$ but the limit from the right is zero) so the limit does not exist.
