Q. 14 - Calculus

Question

Consider the limit expression

$\underset{x\to 0^{+}}{\lim }\frac{\sqrt{x}-x^3}{x^2}$

Calculate the limit.

Step-by-Step Solution

Answer

The limit  of the expression is $\infty$.

Step 1. Given

The given expression is $\underset{x\to 0^{+}}{\lim }\frac{\sqrt{x}-x^3}{x^2}$.

Step 2. Calculation

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