Limit at infinity of cubic roots and square roots without using conjugate $\lim_{x \to \infty} \frac{\sqrt[3]{x+2}}{\sqrt{x+3}}$

Question

$$\lim_{x \to \infty} \frac{\sqrt[3]{x+2}}{\sqrt{x+3}} $$

How would you proceed to find this limit, by eyeballing I would guess it foes to zero since the numerator has a smaller power than the denominator, normaly I would use the binomial theorem if I had something like $$\lim_{x \to \infty} \frac{\sqrt[3]{x+2}-1}{\sqrt{x+3}-1} $$ But here I don't know how to find the limit since I can't really use the binomial theorem.

Answer 1

If you factorize you get $$\frac{x^{1/3}(1+2/x)^{1/3}}{x^{1/2}(1+3/x)^{1/2}} = \frac{(1+2/x)^{1/3}}{x^{1/6}(1+3/x)^{1/2}} $$ I'll let you do the limit yourself.

Answer 2

$$\lim_{x \to \infty} \frac{\sqrt[3]{x+2}}{\sqrt{x+3}}=\lim_{x \to \infty} \sqrt[6]{\dfrac{(x+2)^2}{(x+3)^3}}=\sqrt[6]{0}=0$$

Answer 3

For $x \ge 2$ we have

$0 \le \frac{^3\sqrt{x+2}}{\sqrt{x + 3}} \le \frac{\sqrt[3]{2x}}{\sqrt{x}}$.

Your turn!

Answer 4

Hint:

$$\frac{\sqrt[3]{x+2}}{\sqrt{x+3}}=\frac{\sqrt[3]x}{\sqrt x}\frac{\sqrt[3]{1+\frac2x}}{\sqrt{1+\frac3x}}.$$

Answer 5

Simply use equivalents: $\;\sqrt[3]{x+2}\sim_\infty \sqrt[3]{x}$, $\;\sqrt{x+3}\sim_\infty \sqrt{x}$, hence $$\frac{\sqrt[3]{x+2}}{\sqrt{x+3}}\sim_\infty \frac{x^{1/3}}{x^{1/2}}=x^{-1/6}\xrightarrow[x\to\infty]{}0.$$

Answer 6

A slightly longer way: use Generalized Binomial coefficients: $$ x^{-\frac{1}{6}}\frac{(1+\frac{2}{x})^{\frac{1}{3}}}{(1+\frac{3}{x})^{\frac{1}{2}}} \sim x^{-\frac{1}{6}}\frac{1+\frac{2}{x} + O(x^{-2})}{1+\frac{3}{x} + O(x^{-2}) } \to_x 0 $$

Answer 7

Hint: $x +2 < x + 3$.

(Yes, really, you can solve it using this.)