COROLLARY 5.5. Let f, g : A → R and let c be a limit point of A. Assume that lim f(x) = L and lim g(x) = M. Then: (1) lim [k · f(x)] = k · L for any k E R. %3D (2) lim [f(x) + g(x)] = L+ M. (3) lim [f(x)· g(x)] = L· M. f(x) M' L provided M + 0 and g(x) # 0 for any x E A. (4) lim r→c g(x)
COROLLARY 5.5. Let f, g : A → R and let c be a limit point of A. Assume that lim f(x) = L and lim g(x) = M. Then: (1) lim [k · f(x)] = k · L for any k E R. %3D (2) lim [f(x) + g(x)] = L+ M. (3) lim [f(x)· g(x)] = L· M. f(x) M' L provided M + 0 and g(x) # 0 for any x E A. (4) lim r→c g(x)
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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Please prove (4) under the corollary using the results of theorem 5.3 attached.
Transcribed Image Text:THEOREM 5.3. Suppose that ACR, ƒ : A → R, and c is a limit point of A. Then lim f(x) = L if
and only if, for every sequence (an) C A for which each a, #c and a, → c, we have f(an) → L.
![COROLLARY 5.5. Let f, g : A → R and let c be a limit point of A. Assume that
lim f(x) = L and lim g(x) = M.
Then:
(1) lim [k · f(x)] = k · L for any k E R.
%3D
(2) lim [f(x) + g(x)] = L+ M.
(3) lim [f(x)· g(x)]
= L· M.
f(x)
M'
L
provided M + 0 and g(x) # 0 for any x E A.
(4) lim
r→c g(x)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb708fa5-116d-42c3-bb62-31dd00678e29%2Fc0aa72c3-f4d1-470a-aa52-c547bb81cc1a%2Fto95aa6_processed.png&w=3840&q=75)
Transcribed Image Text:COROLLARY 5.5. Let f, g : A → R and let c be a limit point of A. Assume that
lim f(x) = L and lim g(x) = M.
Then:
(1) lim [k · f(x)] = k · L for any k E R.
%3D
(2) lim [f(x) + g(x)] = L+ M.
(3) lim [f(x)· g(x)]
= L· M.
f(x)
M'
L
provided M + 0 and g(x) # 0 for any x E A.
(4) lim
r→c g(x)
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