Chapter 12, Problem 42CLT

To determine

To find: the limit, and to verify answer using a graphing utility.

Answer to Problem 42CLT

The line $y=0$ is the horizontal asymptote.

Explanation of Solution

Given information:

Given limit

  $\underset{x\to \infty }{\lim }\frac{3-x}{x^2+1}$

Calculation:

This can be evaluated as

  $\begin{array}{c}\underset{x\to \infty }{\lim }\frac{3-x}{x^2+1}=\underset{x\to \infty }{\lim }\frac{\frac{3}{x^2}-\frac{1}{x}}{1+\frac{1}{x^2}}\\ =0\end{array}$

Here the limit exist and limit converges to 0 as it approaches infinity.

Let $f\left(x\right)=\frac{3-x}{x^2+1}$ and plot the function $f\left(x\right)$ .

To calculate the model using the T1-83 graphing calculator follows the steps:

Press Y= and enter the equation

  

Press ZOOM and choose the appropriate zoom function, then plot window is like this.

  

By tracing the curve see that the plot has a horizontal asymptote.

  

There is a horizontal asymptote visible in the graph. Note that the line $y=0$ is the horizontal asymptote.