Answered: 6. Use the e, 8 definition to show that…

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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Numerical Analysis

$**Problem 6**: Use the $ \epsilon, \delta $ definition to show that \[ \lim_{x \to 2} \frac{x+1}{x+3} = \frac{3}{5}. \] **Instructions**: 1. Given a small positive number $ \epsilon $, find a corresponding $ \delta > 0 $ such that whenever $ 0 < |x - 2| < \delta $, it follows that \[ \left| \frac{x+1}{x+3} - \frac{3}{5} \right| < \epsilon. \] 2. Solve the inequality for $ x $ to determine $ \delta $ in terms of $ \epsilon $. 3. Verify your solution by confirming that $ \delta $ works for specific values of $ \epsilon $. Note: The concept involves making the expression $ \frac{x+1}{x+3} $ arbitrarily close to $ \frac{3}{5} $ by restricting how close $ x $ is to 2.$

Transcribed Image Text:**Problem 6**: Use the $ \epsilon, \delta $ definition to show that \[ \lim_{x \to 2} \frac{x+1}{x+3} = \frac{3}{5}. \] **Instructions**: 1. Given a small positive number $ \epsilon $, find a corresponding $ \delta > 0 $ such that whenever $ 0 < |x - 2| < \delta $, it follows that \[ \left| \frac{x+1}{x+3} - \frac{3}{5} \right| < \epsilon. \] 2. Solve the inequality for $ x $ to determine $ \delta $ in terms of $ \epsilon $. 3. Verify your solution by confirming that $ \delta $ works for specific values of $ \epsilon $. Note: The concept involves making the expression $ \frac{x+1}{x+3} $ arbitrarily close to $ \frac{3}{5} $ by restricting how close $ x $ is to 2.

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