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(a) Write down a corresponding definition for f(x) to converge to L as x→-00. Then, use the above definition and the definition you wrote to prove that 1 lim 1+00 1 + x² (b) Suppose f: R → R satisfies 1 = lim x-∞ 1 + x² = 0

(a) Write down a corresponding definition for f(x) to converge to L as x→-00. Then, use the above definition and the definition you wrote to prove that 1 lim 1+00 1 + x² (b) Suppose f: R → R satisfies 1 = lim x-∞ 1 + x² = 0

Advanced Engineering Mathematics
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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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 e > 0, there exists some MER such that for all x ≥ M, I |f(x) - L < € In this case, we write f(x) → L as x → ∞, or lim f(x):= L x→∞ If f does not converge to any LER as x → ∞o, we say f diverges as x → ∞o. (a) Write down a corresponding definition for f(x) to converge to L as x→ −x Then, use the above definition and the definition you wrote to prove that 1 lim 1+00 1 + x² (b) Suppose f: R → R satisfies lim f(x) = lim f(x) = L 00+I 1 = lim x-x 1 + x² g(y): E ∞0-+* for some LE R. Define a function g: R → R by [f(1/y) y #0 L y=0 Show that g is continuous at 0. (Hint: For the - definition of continuity, show that you can break y < d into three cases: y = 0, 1/y > 1/6, 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) = lim g(y) = f(0) y-∞0
Transcribed Image Text: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 e > 0, there exists some MER such that for all x ≥ M, I |f(x) - L < € In this case, we write f(x) → L as x → ∞, or lim f(x):= L x→∞ If f does not converge to any LER as x → ∞o, we say f diverges as x → ∞o. (a) Write down a corresponding definition for f(x) to converge to L as x→ −x Then, use the above definition and the definition you wrote to prove that 1 lim 1+00 1 + x² (b) Suppose f: R → R satisfies lim f(x) = lim f(x) = L 00+I 1 = lim x-x 1 + x² g(y): E ∞0-+* for some LE R. Define a function g: R → R by [f(1/y) y #0 L y=0 Show that g is continuous at 0. (Hint: For the - definition of continuity, show that you can break y < d into three cases: y = 0, 1/y > 1/6, 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) = lim g(y) = f(0) y-∞0
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