Limit of $f(x)$ as $x \to 0^{+}$

Question

Find $$\lim_{x \to 0^{+}} \frac{\sqrt{x}-\sqrt[3]{x}}{\sqrt[5]{x}-\sqrt[7]{x}}$$ I tried to apply l'hospital's rule but it didn't work . Multiplying by conjugates also didn't help .

Answer 1

The hint.

Let $x=t^{210}$.

Hence, we need to calculate $$\lim_{t\rightarrow0^+}\frac{t^{105}-t^{70}}{t^{42}-t^{30}},$$ which is $$\lim_{t\rightarrow0^+}\frac{t^{70}\left(t^{35}-1\right)}{t^{30}\left(t^{12}-1\right)}=\lim_{t\rightarrow0^+}\frac{t^{40}\left(t^{35}-1\right)}{t^{12}-1}=\frac{0(0-1)}{0-1}=0.$$

Answer 2

$$f(x)=\frac{x^{1/2}-x^{1/3}}{x^{1/5}-x^{1/7}} =x^{4/21}\frac{x^{1/6}-1}{x^{2/35}-1}.$$ Each term here has a nice limit as $x\to0^+$.

Answer 3

Short answer:

When going to zero, the smallest exponents dominate and your expression is asymptotic to $x^{1/3-1/7}$, hence the limit is $0$.