Q. 74 - 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.

$f\left(x\right)=\left\{\begin{array}{l}\frac{3x^2}{4+x}, if x<2\\ x^2-2, if x\ge 2\end{array}\right.$

Step-by-Step Solution

Answer

The interval on which the function is continuous is 2

Step 1. Given information

The given information is $f\left(x\right)=\left\{\begin{array}{l}\frac{3x^2}{4+x}, if x<2\\ x^2-2, if x\ge 2\end{array}\right.$

Step 2. Calculate the left and right limits

$$\begin{gathered} \underset{x\to 2^{-}}{\lim }f\left(x\right)=\underset{x\to 2^{-}}{\lim }\left(\frac{3x^2}{4+x}\right) \\ =\left(\frac{3{\left(2\right)}^2}{4+2}\right) \\ =\frac{12}{6} \\ =2 \end{gathered}$$

$$\begin{gathered} \underset{x\to 2^{+}}{\lim }f\left(x\right)=\underset{x\to 2^{+}}{\lim }\left(x^2-2\right) \\ =\left(2^2-2\right) \\ =2 \end{gathered}$$

since the left and right limits both exists and are equal to 2 

therefore f is continuous at 2

Step 3. The solution

The interval on which function is continuous is 2