Question

what it means, in terms of limits, for a function to have a removable discontinuity, a jump discontinuity, or an infinite discontinuity at x = c

Short answer

If limit$\underset{x\to c}{\lim }f\left(x\right)\ne f\left(c\right)$then the discontinuity is known as removable discontinuity.

If limit $\underset{x\to c^{-}}{\lim }f\left(x\right)\ne \underset{x\to c^{+}}{\lim }f\left(x\right)$then the discontinuity is known as jump discontinuity.

If limit $\underset{x\to c^{-}}{\lim }f\left(x\right)=\infty or\underset{x\to c^{+}}{\lim }f\left(x\right)=\infty$then the discontinuity is known as infinite discontinuity.

Step-by-step solution

Step 1. Given information. 

A function has a removable discontinuity, a jump discontinuity, or an infinite discontinuity at $x=c.$

Step 2. removable discontinuity.

A function is discontinuous at $x=c$if its limits as role="math" localid="1648280454491" $x\to c$is not equal to the function value at $x=c$and this type of discontinuity is known as removable discontinuity.

$\underset{x\to c}{\lim }f\left(x\right)\ne f\left(c\right)$

Step 3. Jump discontinuity.

A function is discontinuous if its left limit and right limit are not equal and this type of discontinuity is known as jump discontinuity.

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

Step 4. infinite discontinuity.

A function is discontinuous if its graph has vertical or horizontal asymptotes so that its left limit or right limit or both is equal to infinity.

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