Question

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

$f\left(x\right)=\left\{\begin{array}{l}{x}^{2}if\text{x}\le \text{2}\\ 6-xifx>2\end{array}\right.$

$\left(a\right)\underset{x\to 2^{-}}{\lim }f\left(x\right)\left(b\right)\underset{x\to 2^{+}}{\lim }f\left(x\right)\left(c\right)\underset{x\to 2}{\lim }f\left(x\right)$

Short answer

Value of the limit of the function is$4$ .

Step-by-step solution

Step 1. Given information.

The limit of functions here given are,

$f\left(x\right)=\left\{\begin{array}{l}{x}^{2}if\text{x}\le \text{2}\\ 6-xifx>2\end{array}\right.$

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$2$from the left, and also they approach$4$as$x$approaches$2$from the right.

Thus, the left- and right-hand limits are equal.

So we have,

$\underset{x\to 2^{-}}{\lim }f\left(x\right)=4$and $\underset{x\to 2^{+}}{\lim }f\left(x\right)=4$

Therefore, the limits exist and value of,$\underset{x\to 2}{\lim }f\left(x\right)=4$