Chapter 11.1, Problem 10E

To determine

To calculate: The table and use the result to estimate the limit $\underset{x\to -2}{\lim }\frac{x+2}{x^2-4}$

Answer to Problem 10E

The value of the limits is $\underset{x\to -2}{\lim }\frac{x+2}{x^2-4}=-0.25$

Explanation of Solution

Given information:

Given expression of limit as $\underset{x\to -2}{\lim }\frac{x+2}{x^2-4}$ and the table as,

  $\begin{array}{cccccccc}x& -2.1& -2.01& -2.001& -2& -1.999& -1.99& -1.9\\ f\left(x\right)& & & & ?& & & \end{array}$

Let, $f\left(x\right)=\frac{x+2}{x^2-4}$

The value of the function at $x=-2.1$ is,

  $\begin{array}{c}f\left(-2.1\right)=\frac{-2.1+2}{{\left(-2.1\right)}^2-4}\\ \frac{-0.1}{4.41-4}\\ =-\frac{0.1}{0.41}\\ =-0.2439\end{array}$

The value of the function at $x=-2.01$ is,

  $\begin{array}{c}f\left(-2.01\right)=\frac{-2.01+2}{{\left(-2.01\right)}^2-4}\\ =\frac{-0.01}{4.0401-4}\\ =-\frac{0.01}{0.0401}\\ =-0.2494\end{array}$

The value of the function at $x=-2.001$ is,

  $\begin{array}{c}f\left(-2.001\right)=\frac{-2.001+2}{{\left(-2.001\right)}^2-4}\\ \frac{-0.001}{4.004001-4}\\ =-\frac{0.001}{0.004001}\\ =-0.2499\end{array}$

The value of the function at $x=-2$ is,

  $\begin{array}{c}f\left(-2\right)=\frac{-2+2}{{\left(-2\right)}^2-4}\\ =\frac{0}{4-4}\\ =\frac{0}{0}\\ =\text{undefined}\end{array}$

The value of the function at $x=-1.999$ is,

  $\begin{array}{c}f\left(-1.999\right)=\frac{-1.999+2}{{\left(-1.999\right)}^2-4}\\ =\frac{0.001}{-0.003999}\\ =-0.2501\end{array}$

The value of the function at $x=-1.99$ is,

  $\begin{array}{c}f\left(-1.99\right)=\frac{-1.99+2}{{\left(-1.99\right)}^2-4}\\ =\frac{0.01}{-0.0399}\\ =-0.2506\end{array}$

The value of the function at $x=-1.9$ is,

  $\begin{array}{c}f\left(-1.9\right)=\frac{-1.9+2}{{\left(-1.9\right)}^2-4}\\ =\frac{0.1}{-0.39}\\ =-0.2564\end{array}$

Hence, the complete table is,

  $\begin{array}{cccccccc}x& -2.1& -2.01& -2.001& -2& -1.999& -1.99& -1.9\\ f\left(x\right)& -0.2439& -0.2494& -0.2499& \text{undefined}& -0.2501& -0.2506& -0.2564\end{array}$

Therefore, the value of the limits is $\underset{x\to -2}{\lim }\frac{x+2}{x^2-4}=-0.25$