Chapter 11, Problem 10CT

To determine

Verification and result of the given limit $L=\underset{x\to \infty }{\lim }\frac{6}{5x-1}$ using graphing utility.

Answer to Problem 10CT

The limit $L=\underset{x\to \infty }{\lim }\frac{6}{5x-1}$ exists and equal to 0.

Explanation of Solution

Given data: $y$

The given limit is $L=\underset{x\to \infty }{\lim }\frac{6}{5x-1}$ .

Method/Formula used:

The limit $\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)$ provided it exists, and $\underset{x\to a}{\lim }f\left(x\right)=\underset{y\to 0}{\lim }g\left(y\right)$ where $y=\frac{1}{x}$ .

Calculation:

The given limit is $L=\underset{x\to \infty }{\lim }\frac{6}{5x-1}$ . Let $y=\frac{1}{x}$ then $L=\underset{y\to 0}{\lim }\frac{6}{5\frac{1}{y}-1}$ .

Thus, $L=\underset{y\to 0}{\lim }\frac{6y}{5-y}$

  $L=\frac{6\times 0}{5-0}=0$ .

Thus, the given limit exists and $L=0$ .

Verification of limit L with Graph:

The graph of the function $f\left(x\right)=\frac{6}{5x-1}$ is shown in Figure-1 here.

  

From the graph of function $f\left(x\right)=\frac{6}{5x-1}$ , we see that when $x\to \infty$ the graph of function overlap with x-axis. That means graph of $f\left(x\right)$ intersect x-axis at infinity. Or we say when $x\to \infty$ , then $f\left(x\right)=0$ .

Thus, limit of the function exists and equal to 0.