One-sided and two-sided limit of a piecewise function at a boundary point

For $x < 2$, $w(x) = 2x+1$, which is a polynomial and hence continuous, so by direct substitution and the sum and product rules, $\lim_{x \to 2^-} w(x) = 2(2)+1 = 5$.

For $x \ge 2$, $w(x) = x^2$, and by the power rule, $\lim_{x \to 2^+} w(x) = 2^2 = 4$. Since the left-hand limit (5) and right-hand limit (4) are not equal, the two-sided limit $\lim_{x \to 2} w(x)$ does not exist.