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.

It is rather unpopular to use the standard limit $$\lim_{x\to a} \frac{x^n-a^n} {x-a} =na^{n-1}$$ for evaluating limits of algebraic functions, but it is my preferred approach.
Dividing the numerator and denominator of the given expression by $x-3$ and putting $x+5=u$ in numerator and $v=x-2$ in denominator we can see that the desired limit is equal to $$\dfrac{{\displaystyle \lim_{u\to 8}\dfrac{u^{1/3}-8^{1/3}} {u-8}}} {{\displaystyle \lim_{v\to 1}\dfrac{v^{1/4}-1}{v-1}}}=\dfrac{\dfrac{1}{3}\cdot 8^{-2/3}}{\dfrac{1}{4}\cdot 1^{-3/4}} =\frac{1}{3}$$
