Question

$\underset{x\to 27}{\lim }\frac{\left(x^{1/3}+3\right)\left(x^{1/3}-3\right)}{x-27}$

Answer 1

$$\begin{gathered} \underset{x\to 27}{\lim }\frac{\left[x^{{\displaystyle \frac{1}{3}}}+3\right]\left[x^{{\displaystyle \frac{1}{3}}}-3\right]}{x-27} \\ =\underset{x\to 27}{\lim }\left[\frac{\left(x^{{\displaystyle \frac{1}{3}}}+3\right)\left(x^{{\displaystyle \frac{1}{3}}}-3\right)}{{\left(x^{{\displaystyle \frac{1}{3}}}\right)}^3-3^3}\right] \\ x\to 27 \\ \therefore x^{\frac{1}{3}}\to 3 \\ \mathrm{Let}y=x^{\frac{1}{3}} \\ \underset{y\to 3}{\lim }\left[\frac{\left(y+3\right)\left(y-3\right)}{y^3-3^3}\right] \\ =\frac{\left(3+3\right)}{3\times 3^{3-1}} \\ =\frac{6}{3\times 9} \\ =\frac{2}{9} \end{gathered}$$