Determine whether each of the two attached limit functions are continuous at specific points with a formal proof using mathematical induction. There are three definitions from the text for reference. Thank you! Definition 3.2.1. Let ICR be an open interval, let e E 1, let f: 1– {c} → R be afunction and let L E R. The number L is the limit of f as x goes to c, writtenlim f(x) = L.,if for each e > 0, there is some 8 > 0 such that x € /- {c} and |x – c| < 8 implyf(x) – L| < E. If limf(x) = L, we also say that f converges to L as x goes to c. If fconverges to some real number as x goes to c, we say that lim f(x) exists.ADefinition 3.3.1. Let ACR be a set, and let f: A–R be a function.1. Let c E A. The function f is continuous at c if for each ɛ > 0, there is some8 > 0 such that x E A and |x– c| < 8 imply |f(x) – f(e) < ɛ. The function fis discontinuous at c if f is not continuous at c; in that case we also say thatf has a discontinuity at c.2. The function f is continuous if it is continuous at every number in A. Thefunction f is discontinuous if it is not continuous.ADefinition 3.4.1. Let A CR be a set, and let f: A – R be a function. The function fis uniformly continuous if for each ɛ > 0, there is some 8 > 0 such that x, y € A and|r- y| < 8 imply |f(x) – f(y)| < ɛ. limx-1 x - 1(--89x ) such that x € Rlim (x2 – 1) such that x E Rx-3

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Answered: 8 lim x-1 x-1 (76 - 9x) such that x ER…

Math

Advanced Math

8 lim x-1 x-1 (76 - 9x) such that x ER lim(x² – 1) such that x € R x-3

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Determine whether each of the two attached limit functions are continuous at specific points with a formal proof using mathematical induction. There are three definitions from the text for reference. Thank you!

Definition 3.2.1. Let ICR be an open interval, let e E 1, let f: 1– {c} → R be a
function and let L E R. The number L is the limit of f as x goes to c, written
lim f(x) = L.,
if for each e > 0, there is some 8 > 0 such that x € /- {c} and |x – c| < 8 imply
f(x) – L| < E. If limf(x) = L, we also say that f converges to L as x goes to c. If f
converges to some real number as x goes to c, we say that lim f(x) exists.
A
Definition 3.3.1. Let ACR be a set, and let f: A–R be a function.
1. Let c E A. The function f is continuous at c if for each ɛ > 0, there is some
8 > 0 such that x E A and |x– c| < 8 imply |f(x) – f(e) < ɛ. The function f
is discontinuous at c if f is not continuous at c; in that case we also say that
f has a discontinuity at c.
2. The function f is continuous if it is continuous at every number in A. The
function f is discontinuous if it is not continuous.
A
Definition 3.4.1. Let A CR be a set, and let f: A – R be a function. The function f
is uniformly continuous if for each ɛ > 0, there is some 8 > 0 such that x, y € A and
|r- y| < 8 imply |f(x) – f(y)| < ɛ.

lim
x-1 x - 1
(--
8
9x ) such that x € R
lim (x2 – 1) such that x E R
x-3

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