10. Consider the following theorem, referenced as Theorem 1. Let f: D R, g:D-R, and let a be an accumulation point of D. If f(x) = 9(x) for 1 # a and lim, a 9(1) = L, then lim, a f(r) exists and lim f(r) = lim g(x). a Do the following where Vī - V3 I - 3 1 g(x) = VI+ V3 f(x) = h(z) = /I (a) Use the fact that h is continuous on (0, 00) to compute lim,a VI. (b) Show that f(x) = g(x) if x # 3. (c) Use Theorem 1 above, limit properties, and the result in (a) to compute the following limit. Show explicitly where these known facts are used. Vī – V3 lim 3 I-3
10. Consider the following theorem, referenced as Theorem 1. Let f: D R, g:D-R, and let a be an accumulation point of D. If f(x) = 9(x) for 1 # a and lim, a 9(1) = L, then lim, a f(r) exists and lim f(r) = lim g(x). a Do the following where Vī - V3 I - 3 1 g(x) = VI+ V3 f(x) = h(z) = /I (a) Use the fact that h is continuous on (0, 00) to compute lim,a VI. (b) Show that f(x) = g(x) if x # 3. (c) Use Theorem 1 above, limit properties, and the result in (a) to compute the following limit. Show explicitly where these known facts are used. Vī – V3 lim 3 I-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
10
Transcribed Image Text:10. Consider the following theorem, referenced as Theorem 1.
Let f : D -R, g : D → R, and let a be an accumulation point of D.
If f(x) = g(x) for r # a and lim, a 9(x) = L, then lim, a f(x) exists and
lim f(x) = lim g(r).
Do the following where
VI – V3
1
f(x) =
g(r) =
h(x) = VT
r-3
VI+ V3
(a) Use the fact that h is continuous on (0, 0) to compute lim,-a r.
(b) Show that f(x) = g(x) if x #3.
(c) Use Theorem 1 above, limit properties, and the result in (a) to compute the
following limit. Show explicitly where these known facts are used.
VE - V3
lim
I-3
I-3
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