Question

${lim}_{x\to 27}\frac{(x^{\frac{1}{3}}+3)(x^{\frac{1}{3}}-3)}{x-27}$

Answer 1

${lim}_{x\to 27}\frac{(x^{\frac{1}{3}}+3)(x^{\frac{1}{3}}-3)}{x-27}$

$={lim}_{x\to 27}\frac{(x^{\frac{2}{3}}-9)}{x-27}$

$={lim}_{x\to 27}\frac{x^{\frac{2}{3}}-{27}^{\frac{2}{3}}}{x-27}$

Applying formula $={lim}_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$

$=\frac{2}{3}(27{)}^{\frac{2}{3}-1}$

$=\frac{2}{3}(27{)}^{\frac{-1}{3}}$

$=\frac{2}{3}\times \frac{1}{(27{)}^{\frac{1}{3}}}$

$=\frac{2}{3}\times \frac{1}{3}=\frac{2}{9}$