Q. 12 - Calculus

Question

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.  

$\underset{x\to 2^{-}}{\lim }f\left(x\right)=-\infty , \underset{x\to 2^{+}}{\lim }f\left(x\right)=\infty , f\left(2\right)=3.$

Step-by-Step Solution

Answer

The type of discontinuity is an infinite discontinuity and there is not any one-sided continuity.

The graph of f is 

Step 1. Given Information.

The given function is $\underset{x\to 2^{-}}{\lim }f\left(x\right)=-\infty , \underset{x\to 2^{+}}{\lim }f\left(x\right)=\infty , f\left(2\right)=3.$

Step 2. Describing the discontinuity.

From the function, we can depict that f(x) has infinite discontinuity because both of $\underset{x\to 2^{-}}{\lim }f\left(x\right) \mathrm{and}\underset{\mathrm{x}\to 2^{+}}{\lim }\mathrm{f}\left(\mathrm{x}\right)$are infinite.

Step 3. Describing one-sided continuity at x=c .

There is not any one-sided continuity at $x=2$ because $\underset{x\to 2^{-}}{\lim }\ne f\left(2\right) \mathrm{and} \underset{\mathrm{x}\to 2^{+}}{\lim }\ne \mathrm{f}\left(2\right).$ 

Step 4. Graph of f .

The graph of f is