Calculate this limit : $\lim_{x \to \infty} \frac{\sqrt{x}+\sqrt[4]{x}}{\sqrt[3]{x}+\sqrt[6]{x}}$

Question

Calculate this limit :

$$\lim_{x \to +\infty} \frac{\sqrt{x}+\sqrt[4]{x}}{\sqrt[3]{x}+\sqrt[6]{x}}$$

It is not allowed to use Hopital rule or taylor I tried to factorize but i didn't find any result

Answer 1

Just take $x = y^{12}$. Then, the limit changes to: $$ \lim _{y \to +\infty} \frac{y^6 +y^3}{y^4 + y^2} = \lim _{y \to +\infty} \frac{y^6 }{y^4 + y^2} + \lim _{y \to +\infty} \frac{y^3}{y^4 + y^2} $$

(Edit, thanks to @egreg)It is easily observe the following: $$ \frac{y^6 +y^3}{y^4 + y^2} = y^2 \left( \frac{1 + \frac 1{y^3}}{1 + \frac 1{y^2}}\right) $$

and while the limit of the bracketed portion is $1$, $y^2$ is unbounded, hence we have that the whole term is unbounded.

Answer 2

Hint: Numerator is bigger than denominator