Question
Evaluate the following limits:
$
\lim _{x \rightarrow 2}\left[\frac{x^{-3}-2^{-3}}{x-2}\right]
$
✓
Answer
$
\begin{aligned}
\lim _{x \rightarrow 2} \frac{x^{-3}-2^{-3}}{x-2} & =(-3) \cdot(2)^{-4} \\
& \cdots\left[\because \lim _{x \rightarrow a} \frac{x^n-\mathrm{a}^n}{x-\mathrm{a}}=\mathrm{n} \cdot \mathrm{a}^{n-1}\right] \\
& =-3 \times \frac{1}{2^4} \\
& =\frac{-3}{16}
\end{aligned}
$
\begin{aligned}
\lim _{x \rightarrow 2} \frac{x^{-3}-2^{-3}}{x-2} & =(-3) \cdot(2)^{-4} \\
& \cdots\left[\because \lim _{x \rightarrow a} \frac{x^n-\mathrm{a}^n}{x-\mathrm{a}}=\mathrm{n} \cdot \mathrm{a}^{n-1}\right] \\
& =-3 \times \frac{1}{2^4} \\
& =\frac{-3}{16}
\end{aligned}
$
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