One-sided limits and existence of a two-sided limit at a piecewise breakpoint

For $x$ approaching $0$ from the left, $x<0$ so $w(x) = -x$. As $x \to 0^-$, $-x \to 0$, so $\lim_{x\to 0^-} w(x) = 0$.

For $x$ approaching $0$ from the right, $x \ge 0$ so $w(x) = x^2+1$. As $x \to 0^+$, $x^2+1 \to 1$, so $\lim_{x\to 0^+} w(x) = 1$. Since the left-hand limit ($0$) does not equal the right-hand limit ($1$), the two one-sided limits disagree, so $\lim_{x\to 0} w(x)$ does not exist.