\lim_{x\rightarrow -2^{+}}{\frac{x-2}{x+2}}=\lim_{\varepsilon\rightarrow 0}{\frac{-2+\varepsilon-2}{-2+\varepsilon+2}}=\lim_{\varepsilon\rightarrow 0}{\frac{-4+\varepsilon}{\varepsilon}}=-\infty