Please prove (4) under the corollary using the results of theorem 5.3 attached. THEOREM 5.3. Suppose that ACR, ƒ : A → R, and c is a limit point of A. Then lim f(x) = L ifand 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 thatlim 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'Lprovided M + 0 and g(x) # 0 for any x E A.(4) limr→c g(x)

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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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Question

100%

Please prove (4) under the corollary using the results of theorem 5.3 attached.

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)

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