1. Let's take a look at limits of functions at infinity. Given a function f: R→ R and some LER, we say f(x) converges to L as x → ∞ if for all > 0, there exists some MER such that for all x > M, |f(x) - L| < E In this case, we write f(x) → Las x → ∞, or lim f(x) = L 818 If f does not converge to any LER as x→∞o, we say f diverges as x→∞. (a) Write down a corresponding definition for f(x) to converge to Las x → -∞. Then, use the above definition and the definition you wrote to prove that 1 1 lim = lim xx1+x² 1+x1+x² (b) Suppose f: R→ R satisfies lim f(x)= lim_ f(x) = L 848 for some LE R. Define a function g: R → R by [ƒ(1/y) y‡0 y=0 g(y): = 0 '= Show that g is continuous at 0. (Hint: For the E-d definition of continuity, show that you can break |y| < d into three cases: y = 0, 1/y> 1/8, or -1/y> 1/6. This might help you find the right value of 8.) (c) Continuing from (b), show that if f is continuous at 0, then lim g(y) y4x lim g(y) = f(0) y-1x
1. Let's take a look at limits of functions at infinity. Given a function f: R→ R and some LER, we say f(x) converges to L as x → ∞ if for all > 0, there exists some MER such that for all x > M, |f(x) - L| < E In this case, we write f(x) → Las x → ∞, or lim f(x) = L 818 If f does not converge to any LER as x→∞o, we say f diverges as x→∞. (a) Write down a corresponding definition for f(x) to converge to Las x → -∞. Then, use the above definition and the definition you wrote to prove that 1 1 lim = lim xx1+x² 1+x1+x² (b) Suppose f: R→ R satisfies lim f(x)= lim_ f(x) = L 848 for some LE R. Define a function g: R → R by [ƒ(1/y) y‡0 y=0 g(y): = 0 '= Show that g is continuous at 0. (Hint: For the E-d definition of continuity, show that you can break |y| < d into three cases: y = 0, 1/y> 1/8, or -1/y> 1/6. This might help you find the right value of 8.) (c) Continuing from (b), show that if f is continuous at 0, then lim g(y) y4x lim g(y) = f(0) y-1x
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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Transcribed Image Text:1. Let's take a look at limits of functions at infinity.
Given a function f: R→ R and some LER, we say f(x) converges to L as x → ∞ if
for all > 0, there exists some MER such that for all x > M,
|f(x) - L| < E
In this case, we write f(x) → Las x → ∞, or
lim f(x):= L
818
If f does not converge to any LER as x→∞o, we say f diverges as x→∞.
(a) Write down a corresponding definition for f(x) to converge to Las x → -∞.
Then, use the above definition and the definition you wrote to prove that
1
lim
x+∞ 1+x²
(b) Suppose f: R → R satisfies
lim f(x)= lim_ f(x) = L
#48
2118
g(y):
1
lim
1+-∞ 1+x²
for some LE R. Define a function g: R → R by
[ƒ(1/y) y‡0
y=0
'=
= 0
lim g(y)
y-x
Show that g is continuous at 0.
(Hint: For the E-d definition of continuity, show that you can break |y| < d into
three cases: y = 0, 1/y> 1/8, or -1/y> 1/6. This might help you find the right
value of 8.)
(c) Continuing from (b), show that if f is continuous at 0, then
lim g(y) = f(0)
y--x
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