Continuity of Piecewise Function | Calculus

Question:

Consider $$f(x)=\{\begin{array}{ll}\frac{x^2-x-6}{x+2}& x\ne -2\\ -5& x=-2\end{array}$$ Check whether the function $f(x)$ is continuous at $x=-2$.

Answer:

$$f(-2)=5$$ $$\underset{x\to -2}{lim}f(x)=\underset{x\to -2}{lim}\frac{x^2-x-6}{x+2}$$ $$=\underset{x\to -2}{lim}f(x)=\underset{x\to -2}{lim}\frac{(x-3)(x+2)}{x+2}$$ $$=\underset{x\to -2}{lim}(x-3)=f(-2)=-5$$ Hence $f(x)$ is continuous at $x=-2$