Q. 53 - Calculus

Question

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

Step-by-Step Solution

Answer

The limit is 0

Step 1. Given information

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

Step 2. Calculation

The limit is calculated as below,

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