Chapter 12, Problem 50RE

To determine

To find the limit (if it exists) by evaluating the corresponding one-sided limits.

Answer to Problem 50RE

The limit does not exist.

Explanation of Solution

Given information:

Graph the function. Determine the limit (if it exists) by evaluating the corresponding one-sided limits.

  $\underset{x\to 0}{\lim }f\left(x\right)wheref\left(x\right)=\{\begin{array}{l}x-6,x\ge 0\\ x^2-4,x<0\end{array}$

Calculation:

The limit of given function is,

  $\underset{x\to 0}{\lim }f\left(x\right)wheref\left(x\right)=\{\begin{array}{l}x-6,x\ge 0\\ x^2-4,x<0\end{array}$

Let plot the given function $f\left(x\right)$ .

To calculate the model graphically use graphic utility TI-83calculator,

Press zoom and choose appropriate zoom function, then plot the window is as,

  

Press Trace and give $x$ value at which $y$ value need to be found.

  

From the graph we observe that the function reaches to $-4$ as it approaches $0$ from right hand side and $-6$ from left hand side.

Therefore, the one sided limit is,

  $\begin{array}{l}\underset{x\to 0}{\lim }x-6=-6\\ \underset{x\to 0}{\lim }{x}^2-4=-4\end{array}$

Hence the limit does not exist.