Question

Graph the piecewise-defined function and use your graph to find the values of the limits, if they exist.

$f\left(x\right)=\left\{\begin{array}{l}-x\text{+3}if\text{x}<-1\\ \text{3}ifx\ge -1\end{array}\right.$

$\left(a\right)\underset{x\to -1^{-}}{\lim }f\left(x\right)\left(b\right)\underset{x\to -1^{+}}{\lim }f\left(x\right)\left(c\right)\underset{x\to -1}{\lim }f\left(x\right)$

Short answer

Limit doesn’t exist.

Step-by-step solution

Step 1. Given information.

The limit of functions here given are,

$f\left(x\right)=\left\{\begin{array}{l}-x\text{+3}if\text{x}<-1\\ \text{3}ifx\ge -1\end{array}\right.$

Step 2. Graph.

The graph of $f$is shown in the figure.

Step 3. Details and conclusion.

From the graph, we see that the values of $f\left(x\right)$approach $4$as $x$approaches $-1$from the left, but they approach $3$as $x$approaches $-1$from the right.

Thus, the left- and right-hand limits are not equal.

So we have,

$\underset{x\to -1^{-}}{\lim }f\left(x\right)=4$and$\underset{x\to -1^{+}}{\lim }f\left(x\right)=3$

Therefore, $\underset{x\to -1}{\lim }f\left(x\right)$doesn’t exist.