Chapter 12, Problem 40CLT

To determine

Find the limit (if it exists).

Answer to Problem 40CLT

Limit does not exist.

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.

  $\underset{x\to \infty }{\lim }\frac{x^3}{x^2-9}$

Calculation:

Consider the given limit.

  $\underset{x\to \infty }{\lim }\frac{x^3}{x^2-9}$

Now evaluate the limit as follows,

  $\underset{x\to \infty }{\lim }\frac{x^3}{x^2-9}=\underset{x\to \infty }{\lim }\frac{x}{1-\frac{9}{x^2}}$

The limit does not exist because the numerator decreases without bound as the denominator approaches $1$ .

Now function $\underset{x\to \infty }{\lim }\frac{x^3}{x^2-9}$ and plot the by using $TI-83$ graphic calculator,

Press Y= and enter the function.

  

Press zoom and plot the window.

  

The plot does not have horizontal asymptote.

Hence plot does not have horizontal asymptote so limit does not exist.