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Q. 9

Expert-verified
Found in: Page 777

Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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

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

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

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

See the 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.

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

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.

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

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.

$\underset{x\to {c}^{-}}{\mathrm{lim}}f\left(x\right)=\infty \phantom{\rule{0ex}{0ex}}or\phantom{\rule{0ex}{0ex}}\underset{x\to {c}^{+}}{\mathrm{lim}}f\left(x\right)=\infty$