Question

Evaluate: $\underset{x\to 1}{lim}\frac{2(x-1)}{\sqrt[3]{3(7+x)}-2}$

Answer 1

$D^r=p^{1/3}-q^{1/3},(p=(7+x{)}^{1/3},q=8).$
Rationalize by multiplying $N^r$ and $D^r$ by $p^{2/3}+q^{2/3}+{\left(pq\right)}^{1/3}$

$\underset{x\to 1}{lim}\frac{2(x-1)}{7+x-8}\times {(7+x)}^{2/3}+8^{2/3}+{\left\{8(7+x)\right\}}^{1/3}$

$=2[8^{2/3}+8^{2/3}+{\left(64\right)}^{1/3}]=2\left[12\right]=24$