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?

For example
$$f(x):=\frac{|x|}x\implies\;\lim_{x\to 0}f(x)=\;\text{does not exist (check one sided limits), but}$$
$$\lim_{x\to 0}f(x^2)=\lim_{x\to 0}\frac{x^2}{x^2}=1\;\;\text{does exist}\;\ldots$$
The first part above is nothing else but the proof of the well known fact that the absolute value function isn't differentiable at zero.
