Question
Find the limit: $\lim \limits_{x \rightarrow 2}\left[\frac{x^{2}-4}{x^{3}-4 x^{2}+4 x}\right]$

Answer

Evaluating the function at 2, we get it of the form $\frac{0}{0}$
Therefore, we have,
$\lim _{x \rightarrow 2} \frac{x^{2}-4}{x^{3}-4 x^{2}+4 x}=\lim _{x \rightarrow 2} \frac{(x+2)(x-2)}{x(x-2)^{2}}$
$=\lim _{x \rightarrow 2} \frac{(x+2)}{x(x-2)}=\frac{2+2}{2(2-2)}=\frac{4}{0}$ = $\infty$

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