Q. 54 - Calculus

Question

Calculate each limit in Exercises 35–80.
$\underset{x\to 0^{+}}{lim}\frac{x^{-3}}{x^2-x^{-1}}$

Step-by-Step Solution

Answer

The limit is infinity.

Step 1. Given information

The given expression is  $\underset{x\to 0^{+}}{lim}\frac{x^{-3}}{x^2-x^{-1}}$

Step 2. Calculation

The limit is calculated as below, 

$$\begin{gathered} \underset{x\to 0^{+}}{lim}\frac{x^{-3}}{x^2-x^{-1}}=\underset{x\to 0^{+}}{lim}\frac{{\displaystyle \frac{1}{x^3}}}{x^2-{\displaystyle \frac{1}{x}}} \\ =\underset{x\to 0^{+}}{lim}\frac{{\displaystyle \frac{1}{x^3}}}{{\displaystyle \frac{x^3-1}{x}}} \\ =\underset{x\to 0^{+}}{lim}\frac{1}{x^2(x^3-1)} \\ =\frac{1}{0^2(0^3-1)} \\ =\infty \end{gathered}$$