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Problem 1. The following function has domain (-1,0) U (0, ∞), −1+√1+x X f(x) = Make a table of values of f(x) for x = ±1, 0.1, 0.01, and ±0.001. Use your table to guess a limiting value for f(x) as x→0. Finally, use difference- of-squares rationalization to find a continuous function g(x) whose domain contains (-1, ∞) such that f(x) equals g(x) on (-1,0) U (0, ∞). Use this to prove that your guess is correct.

Problem 1. The following function has domain (-1,0) U (0, ∞), −1+√1+x X f(x) = Make a table of values of f(x) for x = ±1, 0.1, 0.01, and ±0.001. Use your table to guess a limiting value for f(x) as x→0. Finally, use difference- of-squares rationalization to find a continuous function g(x) whose domain contains (-1, ∞) such that f(x) equals g(x) on (-1,0) U (0, ∞). Use this to prove that your guess is correct.

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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Problem 1. The following function has domain (−1, 0) ∪ (0, ∞), f(x)=(-1+√1+x)/x

Make a table of values of f (x) for x = ±1, ±0.1, ±0.01, and ±0.001. Use your table to guess a limiting value for f(x) as x → 0. Finally, use difference- of-squares rationalization to find a continuous function g(x) whose domain contains (−1, ∞) such that f (x) equals g(x) on (−1, 0) ∪ (0, ∞). Use this to prove that your guess is correct.

 

Problem 1. The following function has domain (-1,0) U (0, ∞0), −1+√1+x Xx f(x) = Make a table of values of f(x) for x = ±1, 0.1, 0.01, and ±0.001. Use your table to guess a limiting value for f(x) as x → 0. Finally, use difference- of-squares rationalization to find a continuous function g(x) whose domain contains (-1, ∞) such that f(x) equals g(x) on (-1,0) U (0, ∞o). Use this to prove that your guess is correct. Problem 2. Define a function h(x) on (-∞, +∞) by the following rule, sin (1/x), x 0. x=0 h(x) =X{ 0, Compute the limit as x approaches 0 of k(x) = xh(x). Explain why it is invalid to simply compute the limit as k(0) = 0.h(0)=0.0=0? 81 YOUKY
Transcribed Image Text:Problem 1. The following function has domain (-1,0) U (0, ∞0), −1+√1+x Xx f(x) = Make a table of values of f(x) for x = ±1, 0.1, 0.01, and ±0.001. Use your table to guess a limiting value for f(x) as x → 0. Finally, use difference- of-squares rationalization to find a continuous function g(x) whose domain contains (-1, ∞) such that f(x) equals g(x) on (-1,0) U (0, ∞o). Use this to prove that your guess is correct. Problem 2. Define a function h(x) on (-∞, +∞) by the following rule, sin (1/x), x 0. x=0 h(x) =X{ 0, Compute the limit as x approaches 0 of k(x) = xh(x). Explain why it is invalid to simply compute the limit as k(0) = 0.h(0)=0.0=0? 81 YOUKY
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