Evaluating $\lim_{x\to3}\frac{\sqrt[3]{x+5}-2}{\sqrt[4]{x-2}-1}$ without L'Hopital I have the following limit question, where different indices of roots appear in the numerator and the denominator
$$\lim_{x\to3}\frac{\sqrt[3]{x+5}-2}{\sqrt[4]{x-2}-1}.$$
As we not allowed to use L'Hopital, I want to learn how we can proceed algebraically.

By definition of derivative, by $f(x)=\sqrt[3]{x+5}$ and $g(x)=\sqrt[4]{x-2}$ we have
$$\lim_{x\to3}\frac{\sqrt[3]{x+5}-2}{\sqrt[4]{x-2}-1}=\lim_{x\to3}\frac{\sqrt[3]{x+5}-2}{x-3}\frac{x-3}{\sqrt[4]{x-2}-1}=\frac{f'(3)}{g'(3)}$$
