Chapter 12, Problem 43CLT

To determine

Find the limit (if it exists).

Answer to Problem 43CLT

Limit is existing $\boxed{y=0}$

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 your result graphically.

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

Calculation:

Consider the given limit.

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

Now evaluate the limit as follows,

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

The limit exists and converse the value $0$ as is goes to infinity.

Now function $f\left(x\right)=\frac{2x}{x^2+3x-2}$ and plot the by using $TI-83$ graphic calculator,

Press Y= and enter the function.

  

Press zoom and plot the window.

  

By tracing the curve and observe that the plot has a horizontal asymptote.

  

Hence the line $\boxed{y=0}$ is a horizontal asymptote.