Q. 51 - Calculus

Question

Calculate each limit in Exercises

$\underset{x\to 0^{+}}{\lim }\left(x^{-\frac{1}{3}}-x^{-\frac{1}{2}}\right)$

Step-by-Step Solution

Answer

The value of $\underset{x\to 0^{+}}{\lim }\left(x^{-\frac{1}{3}}-x^{-\frac{1}{2}}\right)$is $-\infty .$

Step 1. Given information

The given limit is $\underset{x\to 0^{+}}{\lim }\left(x^{-\frac{1}{3}}-x^{-\frac{1}{2}}\right).$

Step 2. value of the expression.

Evaluate the expression.

$$\begin{gathered} \underset{x\to 0^{+}}{\lim }\left(x^{-\frac{1}{3}}-x^{-\frac{1}{2}}\right)=\underset{x\to 0^{+}}{\lim }\left(\frac{1}{x^{\frac{1}{3}}}-\frac{1}{x^{\frac{1}{2}}}\right) \\ =\underset{x\to 0^{+}}{\lim }\left(\frac{1}{\sqrt[3]{x}}-\frac{1}{\sqrt{x}}\right) \\ =\frac{1}{\sqrt[3]{0}}-\frac{1}{\sqrt{0}} \\ =\infty -\infty \\ =-\infty \end{gathered}$$

The value of limit is $-\infty .$