\[ \begin{split} &\lim_{x \to 2} \frac{x^4-8x^2+16}{(x-2)(x-3)}=\\[16pt] &=\lim_{x \to 2}\frac{(x^2-4)^2}{(x-2)(x-3)}=\\[16pt] &=\lim_{x \to 2}\frac{(x-2)^2(x+2)^2}{(x-2)(x-3)}=\\[16pt] &=\lim_{x \to 2}\frac{(x-2)(x+2)^2}{x-3}=\\[16pt] &=\frac{0}{-1}=0 \end{split} \]