Q. 76 - Calculus

Question

Describe the intervals on which each function f  is continuous. At each point where f fails to be continuous, use limits to determine the type of discontinuity and any left- or right-continuity.

$f\left(x\right)=\left\{\begin{array}{ll}x+1,& if x<3\\ 2,& if x=3\\ x^2-9,& if x>3\end{array}\right.$

Step-by-Step Solution

Answer

The interval on which the function is discontinuous.

Step 1. Given information.

Consider the given question,

$f\left(x\right)=\left\{\begin{array}{ll}x+1,& if x<3\\ 2,& if x=3\\ x^2-9,& if x>3\end{array}\right.$

Step 2. Calculate the left and right limits.

We need to calculate the left and right limits,

$$\begin{gathered} \underset{x\to 3^{+}}{\lim }f\left(x\right)=\underset{x\to 3^{+}}{\lim }\left(x+1\right) \\ =\left(3+1\right) \\ =4 \\ \underset{x\to 3}{\lim }f\left(x\right)=\underset{x\to 3}{\lim }\left(2\right) \\ =2 \\ \underset{x\to 3^{-}}{\lim }f\left(x\right)=\underset{x\to 3^{-}}{\lim }\left(x^2-9\right) \\ =\left[{\left(3\right)}^2-9\right] \\ =9-9 \\ =0 \end{gathered}$$

As the left and right  limits both exists but not equal.

Hence, the function is discontinuous.