Q. 9 - 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. 

$\underset{x\to -1^{-}}{\lim }f\left(x\right)=2, \underset{x\to -1^{+}}{\lim }f\left(x\right)=2, f(-1)=1.$

Step-by-Step Solution

Answer

The type of discontinuity is a removable discontinuity and there is not any one-sided continuity. 

The possible graph of f is

Step 1. Given Information.

The given function is $\underset{x\to -1^{-}}{\lim }f\left(x\right)=2, \underset{x\to -1^{+}}{\lim }f\left(x\right)=2, f(-1)=1.$

Step 2. Describing the discontinuity.

From the function, we can depict that $\underset{x\to -1}{\lim }$ exists but is not equal to $f(-1).$

Thus, f(x) has a removable discontinuity at $x=-1.$

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

There is not any one-sided continuity at $x=c$ because $\underset{x\to -1^{-}}{\lim }\ne f(-1) \mathrm{and} \underset{\mathrm{x}\to -1^{+}}{\lim }\ne \mathrm{f}(-1) .$

Step 4. Graph of f.

The graph of f is