Chapter 12, Problem 79RE

To determine

Evaluate a Limit at Infinity.

Answer to Problem 79RE

the result is verified.

Explanation of Solution

Given information:

Find the limit (if it exists). If the limit does not exist, then explain why. Use a graphing utility to verify

result graphically.

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

Calculation:

Consider the function,

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

Now evaluate the limit at infinity, if it exists.

For the rational number,

  $f\left(x\right)=\frac{N\left(x\right)}{D\left(x\right)}$

Where,

  $\begin{array}{l}N\left(x\right)=a_n{x}^n+\mathrm{......}+a_0\\ D\left(x\right)=b_m{x}^m+\mathrm{......}+b_0\end{array}$

The limit as $x$ approaches positive and negative infinity,

  $\underset{x\to \infty }{\lim }f\left(x\right)=\{\begin{array}{l}0,ifn<m\\ \frac{a_n}{b_m},ifn=m\end{array}$

The limit does not exist if $n>m$ ,

Here the degree of numerator and denominator is equal, so the limit of the function is ratio of the coefficients.

  $\begin{array}{l}\underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\frac{3+x}{3-x}\\ =\frac{1}{-1}\\ =\boxed{-1}\end{array}$

Now verify the result graphically,

By using $TI83$ calculator draw the graph of the function.

Press $Y=$ write the function and then press ENTER key,

  

Press window and adjust $X,Y$ setting,

  

Draw the graph,

  

Change the window size and press GRAPH key,

  

Press TRACE and move the cursor towards increasing $x$ value,

  

From the graph the function $f\left(x\right)$ is nearer to the value $\boxed{-1}$ as $x$ increases towards infinity.

So $\underset{x\to \infty }{\lim }f\left(x\right)=\boxed{-1}$

Hence the result is verified.