Given any function f(x), explain why we can evaluate limx→+∞ƒ[g(x)] by substituting t= g(x) and writing 45. (a) Suppose g(x)→+∞ as x→+∞. lim f[g(x)] = lim f(t) - x41x t→+∞ (Here, "equality" is interpreted to mean that either both limits exist and are equal or that both limits fail to exist.) (b) Why does the result in part (a) remain valid if limx→→+ is replaced everywhere by one of limx→∞, limx→c, limx→c-, or limx→c+?
Given any function f(x), explain why we can evaluate limx→+∞ƒ[g(x)] by substituting t= g(x) and writing 45. (a) Suppose g(x)→+∞ as x→+∞. lim f[g(x)] = lim f(t) - x41x t→+∞ (Here, "equality" is interpreted to mean that either both limits exist and are equal or that both limits fail to exist.) (b) Why does the result in part (a) remain valid if limx→→+ is replaced everywhere by one of limx→∞, limx→c, limx→c-, or limx→c+?
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
![Given any
function f(x), explain why we can evaluate
limx→+ f[g(x)] by substituting t= g(x) and
writing
45. (a) Suppose g(x)→+∞ as x→+∞.
lim f[g(x)] = lim_ f(t)
x→+∞
1→ +∞
(Here, "equality" is interpreted to mean that either
both limits exist and are equal or that both limits
fail to exist.)
(b) Why does the result in part (a) remain valid
if limx→+ is replaced everywhere by one of
limx→→∞, limx→c, limx→c-, or limx→>c+?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba3eb3fc-f971-41fe-b16c-34efe2fbb0c8%2Fc6c02e48-b15e-4911-81b8-47f4f80ad75c%2Feu4tsrq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Given any
function f(x), explain why we can evaluate
limx→+ f[g(x)] by substituting t= g(x) and
writing
45. (a) Suppose g(x)→+∞ as x→+∞.
lim f[g(x)] = lim_ f(t)
x→+∞
1→ +∞
(Here, "equality" is interpreted to mean that either
both limits exist and are equal or that both limits
fail to exist.)
(b) Why does the result in part (a) remain valid
if limx→+ is replaced everywhere by one of
limx→→∞, limx→c, limx→c-, or limx→>c+?
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