Chapter 9.1, Problem 9QR

To determine

To Calculate: The limit of an expression in variable $x$ , when $\text{x}\to \infty$ .

Answer to Problem 9QR

The value of the given limit is $1$

Explanation of Solution

Given Information: The limit to be calculated is $\underset{x\to \infty }{\lim }x\sin \left(\frac{1}{x}\right)$

Concept Used: $\underset{\theta \to 0}{\lim }\frac{\sin \theta }{\theta }=1$

Calculation:In $\underset{x\to \infty }{\lim }xsin\left(\frac{1}{x}\right)$ , replace $\frac{1}{x}$ with $\theta$ to write

  $\underset{x\to \infty }{\lim }xsin\left(\frac{1}{x}\right)=\underset{\frac{1}{x}\to 0}{\lim }\frac{sin\left(\frac{1}{x}\right)}{\left(\frac{1}{x}\right)}$

  $=\underset{\theta \to 0}{\lim }\left(\frac{\sin \theta }{\theta }\right)$

  $=1$

Therefore, the value of the given limit is $1$