Find the limit of $ \lim_{x \to 7} \frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2} $ I need to evaluate the limit without using L'Hopital's rule.
$$\Large
\lim_{x \to 7} \frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}
$$

Letting $x = 7 + h$, then we find: $$\displaystyle \lim_{h \to 0} \dfrac{\sqrt{h+9} - \sqrt[3]{h+27}}{\sqrt[4]{h+16} - 2} = \displaystyle \lim_{h \to 0} \dfrac{3\left(1+\dfrac{h}{18}\right) - 3\left(1+\dfrac{h}{81}\right)}{2\left(1+\dfrac{h}{64}\right) - 2} = \dfrac{112}{27},$$
where $(1+h)^{\alpha} \sim_0 1 + \alpha h$.
