Chapter 11, Problem 48CR

To determine

To graph: The function and evaluate the corresponding one-sided limits. Then determine the limit (if it exists).

Answer to Problem 48CR

  $\underset{\mathrm{x}\to 0}{\lim }\mathrm{f}(\mathrm{x})$ does not exist.

Explanation of Solution

Given information: $\underset{\mathrm{x}\to 0}{\lim }\mathrm{f}(\mathrm{x})$ where $\mathrm{f}(\mathrm{x})=\{\begin{array}{l}\mathrm{x}-6,\mathrm{x}\ge 0\\ {\mathrm{x}}^2-4,\mathrm{x}<0\end{array}$

Formula used: Direct Substitution Method

If by direct substituting of the point in the given expression we get a finite number, then the number obtained is the limit of the given expression.

Graph:

  

Interpretation: From the graph, $\mathrm{f}(\mathrm{x})\ge -6$ for $\mathrm{x}\ge 0$ and $\mathrm{f}(\mathrm{x})\ge -4$ for $\mathrm{x}<0$ .

So, the limit from the left is

  $\underset{\mathrm{x}\to 0^{-}}{\lim }\mathrm{f}(\mathrm{x})=-4$

And the limit from the right is

  $\underset{\mathrm{x}\to 0^{+}}{\lim }\mathrm{f}(\mathrm{x})=-6$

Calculation: Using direct substitution.

For $\mathrm{x}\ge 0$ ,

  $\underset{\mathrm{x}\to 0^{+}}{\lim }\mathrm{f}(\mathrm{x})=\underset{\mathrm{x}\to 0^{+}}{\lim }(\mathrm{x}-6)=0-6=-6$

For $\mathrm{x}<0$ ,

  $\underset{\mathrm{x}\to 0^{-}}{\lim }\mathrm{f}(\mathrm{x})=\underset{\mathrm{x}\to 0^{-}}{\lim }({\mathrm{x}}^2-4)=0-4=-4$

Since one-sided limits are not equal.

Thus, $\underset{\mathrm{x}\to 0}{\lim }\mathrm{f}(\mathrm{x})$ does not exist.