I'm suppose to calculate the derivative of $f(x)=\sqrt{x^2}$ when $x=0$. I.e., I need to determine $f'(0)$. I worked it out this way:
$\begin{align} f'(0 )&= \lim_{x\rightarrow 0} \frac{f(x)-f(a)}{x-a}\\ \\ &=\lim_{x\rightarrow 0} \frac{\sqrt{x^2} - 0}{x-0}\\\\ &=\lim_{x\rightarrow 0} \frac{x}{x}\\ \\ &=1\end{align}$
I know I'm doing something wrong, because the solution says there is no derivative, But I don't know why.
Your issue is the following: $\sqrt{x^{2}}=|x|$, not $x$. So your problem reduces to $$\lim_{x\to 0}\frac{|x|}{x}$$ which does not exist. This is because the right-hand limit ($1$) and left-hand limit $(-1)$ do not agree.