Chapter 13.1, Problem 11E

To determine

To estimate: The value of the limit.

Answer to Problem 11E

The value of the limit is $-1$

Explanation of Solution

Given information:

Limit is $\underset{x\to -4}{\lim }\frac{x+4}{x^2+7x+12}$

Concept used:

Limit is in $\frac{0}{0}$ form it is called indeterminate form then we can use L’ Hospital Rule.

Formula used:

  $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\lim }\frac{f\text{'}\left(x\right)}{g\text{'}\left(x\right)}\text{ when }\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\frac{0}{0}\left(or\right)\frac{\pm \infty }{\pm \infty }$

Calculation:

Limit is $\underset{x\to -4}{\lim }\frac{x+4}{x^2+7x+12}=\frac{-4+4}{16-28+12}=\frac{0}{0}\text{ form}$

Therefore,

  $\begin{array}{l}f\left(1\right)=x+4g\left(x\right)=x^2+7x+12\\ f\text{'}\left(x\right)=1g\left(x\right)=2x+2\\ \underset{x\to -4}{\lim }\frac{x+4}{x^2+7x+12}=\underset{x\to -4}{\lim }\frac{1}{2x+7}=\frac{1}{-8+7}=-1\end{array}$

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression $\underset{x\to -4}{\lim }\frac{x+4}{x^2+7x+12}$ which is required to graph.

Step 4: Press GRAPH button to graph the function and adjust the windows according to the graph.

The graph is obtained as:

  

Verification:

From graph, it can be observed that points near by $x=-4$ y - coordinate is $-1$ nearly.