<s>[INST] 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.
 [/INST] Hint:
Numerator:
$a^3-b^3= (a-b)(a^2+ab+b^2)$.
$a=(x+5)^{1/3}$; $b=2.$
Denominator:
$c^4-d^4=(c^2-d^2)(c^2+d^2)=$
$(c-d)(c+d)(c^2+d^2);$
$c=(x-2)^{1/4};$ $d=1.$
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