1. Let f(x) be a continuous function on [a, b], where a, b = R and a < b. Suppose that there are two sequences (xn) and (yn) satisfying that (a) a
1. Let f(x) be a continuous function on [a, b], where a, b = R and a < b. Suppose that there are two sequences (xn) and (yn) satisfying that (a) a
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
. Let f (x) be a continuous function on [a, b], where a, b ∈ R and a < b. Suppose that there are two sequences
(xn) and (yn) satisfying that
(a) a < xn < c < yn < b for all n ∈ N, and
(b) lim(xn) = c and lim(yn) = c.
REFER TO PICTURE AND PROVE IT, WHILE ALSO EXPLAINING EACH STEP IN FULL DETAIL
![1. Let f(x) be a continuous function on [a, b], where a, b = R and a < b. Suppose that there are two sequences
(xn) and (yn) satisfying that
(a) a <n<c< Yn <b for all n = N, and
(b) lim(xn) =c and lim(yn):
= C.
Prove that if f(x) is differentiable at c, then
f(yn) - f(xn)
lim
=
= f'(c).
n→∞
Yn - xn](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F643c9073-bfc9-41d3-b50c-3b14187fea75%2F0a93fd8a-e600-4445-a1fe-935931725943%2Fd7ulz8l_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let f(x) be a continuous function on [a, b], where a, b = R and a < b. Suppose that there are two sequences
(xn) and (yn) satisfying that
(a) a <n<c< Yn <b for all n = N, and
(b) lim(xn) =c and lim(yn):
= C.
Prove that if f(x) is differentiable at c, then
f(yn) - f(xn)
lim
=
= f'(c).
n→∞
Yn - xn
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