Function $\sqrt[5]{x}$

Question

Wolfram alpha tell me that $\lim_{x\to 0-}\frac{\sqrt[5]{x}}{x}$ does not exists and in particular the domain of $\sqrt[5]{x}$ is given by all $x\geq 0$...why? I should say that $\lim_{x\to 0-}\frac{\sqrt[5]{x}}{x}=+\infty$ and the domain is all $\mathbb{R}$

Answer 1

For $x^{1/5}$, Wolfram|Alpha will take the principal root by default but you can 'force' it to take the real-valued root instead: it then correctly reports the domain to be $\mathbb{R}$.

The same goes for the limit you are trying to calculate: principal (default) vs. real-valued with the correct result: $$\lim_{x \to 0^-}\frac{\sqrt[5]{x}}{x}=+\infty$$

I think it mainly illustrates how you should be very careful with interpreting what mathematical computer software tells you.

See also: Principal value.

Answer 2

Extending StackTD's answer, the first thing to notice is that in the expression below, since $x$ is approaching $0$ from below, both the numerator and the denominator will be negative. Therefore, the fraction is positive.

$$\lim_{x \to 0^-}\frac{\sqrt[5]{x}}{x}.$$

For simplicity, you can therefore convert the problem to evaluating:

$$\lim_{x \to 0^+}\frac{\sqrt[5]{x}}{x}.$$

At this point there are three ways to complete the evaluation, two formal and one informal:

1. divide both the numerator and the denominator by $\sqrt[5]{x}$, which converts the fraction to $\frac{1}{x^{(4/5)}}.$ Here, since the denominator is approaching $0$ from above, the fraction is clearly going to $+\infty.$

2. Apply L'Hopital's Rule, which transforms the fraction into $\frac{[1/5]x^{(-4/5)}}{1}$, which, as in the first approach, clearly evaluates to $+\infty.$

3. Rely solely on intuition. Consider what happens as $x$ progressively takes on the values $0.9, 0.8, 0.7, ...$ so that $x^2$ is simultaneously taking on the values $0.81, 0.64, 0.49, ...$. Clearly as $x$ goes to $0^+$, $x^2$ goes to $0^+$ more quickly (told you this was informal). Therefore, as $x$ goes to $0^+, \sqrt[5]{x}$ goes to $0^+$ more slowly. This insight permits you to reach the accurate conclusion without (much) math.