Chapter 11.1, Problem 9E

To determine

To fill up the table and use the table to estimate the limit numerically.

Answer to Problem 9E

Explanation of Solution

Given:

  $\underset{x\to -2}{\lim }\frac{x+2}{x^2-4}$

x	-2.1	-2.01	-2.001	-2	-1.999	-1.99	-1.9
F(x)

  $\begin{array}{l}\underset{x\to -2}{\lim }\frac{x+2}{x^2-4}\\ \underset{x\to -2}{\lim }\frac{x+2}{(x+2)(x-2)}\\ \underset{x\to -2}{\lim }\frac{1}{x-2}\end{array}$

X	-2.1	-2.01	-2.001	-2	-1.999	-1.99	-1.9
F(x)	-0.24390	-0.24937	-0.24993	-0.25	-0.25006	-0.25062	-0.25641

  $\begin{array}{l}\text{If }x\text{ = -2}\text{.1 then f}\left(x\right)\text{ is -0}\text{.24390}\\ \underset{\text{x}\to -\text{2}}{\text{lim}}\frac{\text{1}}{x\text{-2}}\\ \underset{\text{x}\to -\text{2}}{\text{lim}}\frac{\text{1}}{\text{-2}\text{.1-2}}\end{array}$

  $\begin{array}{l}\underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{\text{-4}\text{.1}}\\ \text{-0}\text{.24390}\\ \text{If }x\text{ = -2}\text{.01 then f}\left(x\right)\text{ is -0}\text{.24937}\end{array}$

  $\begin{array}{l}\underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{x\text{-2}}\\ \underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{\text{-2}\text{.01-2}}\\ \underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{\text{-4}\text{.01}}\\ \text{-0}\text{.24937}\end{array}$

  $\begin{array}{l}\text{ If }x\text{ = -2}\text{.001 then f}\left(x\right)\text{ is -0}\text{.24993}\\ \underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{x\text{-2}}\\ \underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{\text{-2}\text{.001-2}}\end{array}$

  $\begin{array}{l}\underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{\text{-4}\text{.001}}\\ \text{-0}\text{.24993}\\ \text{ If }x\text{ = -2 then f}\left(x\right)\text{ is -0}\text{.25}\end{array}$

  $\begin{array}{l}\underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{x\text{-2}}\\ \underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{\text{-2-2}}\\ \underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{\text{-4}}\end{array}$

  $\begin{array}{l}\text{-0}\text{.25}\\ \text{If }x\text{ = -1}\text{.999 then f}\left(x\right)\text{ is -0}\text{.25006 }\\ \underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{x\text{-2}}\\ \underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{\text{-1}\text{.999-2}}\end{array}$

  $\begin{array}{l}\underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{\text{-3}\text{.999}}\\ \text{-0}\text{.25006}\\ \text{ If }x\text{ = 1}\text{.99 then f}\left(x\right)\text{ is -0}\text{.25062 }\end{array}$

  $\begin{array}{l}\underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{x\text{-2}}\\ \underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{\text{-1}\text{.99-2}}\\ \underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{\text{-3}\text{.99}}\\ \text{-0}\text{.25062}\end{array}$

  $\begin{array}{l}\text{If }x\text{ = 1}\text{.9 then f}\left(x\right)\text{ is-0}\text{.25641}\\ \underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{x\text{-2}}\end{array}$

  $\begin{array}{l}\underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{\text{-1}\text{.9-2}}\\ \underset{\text{x}\to \text{-2}}{\text{lim}}\frac{\text{1}}{\text{-3}\text{.9}}\\ \text{-0}\text{.25641}\end{array}$

Conclusion :

After inspecting this table, we see that the functional values less than -2 appear to be decreasing toward- 0.25 whereas the functional values greater than -2 appear to be increasing toward -0.25. It conclude that $\underset{x\to -2}{\lim }\frac{x+2}{x^2-4}=-0.25$