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Now consider f(x) = ! defined for all real numbers except for x = 0. Appealing to the e-8 definition, prove that no real number LER can be a limit of f(x) as x tends to 0, i.e., no matter what you choose for L, limz0 L. (3.1.4) Preliminary Work Wuit

Now consider f(x) = ! defined for all real numbers except for x = 0. Appealing to the e-8 definition, prove that no real number LER can be a limit of f(x) as x tends to 0, i.e., no matter what you choose for L, limz0 L. (3.1.4) Preliminary Work Wuit

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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**Preliminary Work and Exploration of Limits** **Function Analysis:** Consider the function \( f(x) = \frac{1}{x} \), which is defined for all real numbers except for \( x = 0 \). **Objective:** Prove that no real number \( L \in \mathbb{R} \) can be the limit of \( f(x) \) as \( x \) approaches 0. This is expressed formally as \( \lim_{x \to 0} \frac{1}{x} \neq L \). **Preliminary Work (3.1.4):** - Write down and explore your thought process. - Analyze how \( f(x) \) behaves as \( x \) approaches 0 from both the positive and negative sides. - Use the \( \epsilon\)–\( \delta \) definition of a limit to support your explanation. **Formal Proof (3.1.5):** Prepare a structured and rigorous mathematical proof to demonstrate the claim.
Transcribed Image Text:**Preliminary Work and Exploration of Limits** **Function Analysis:** Consider the function \( f(x) = \frac{1}{x} \), which is defined for all real numbers except for \( x = 0 \). **Objective:** Prove that no real number \( L \in \mathbb{R} \) can be the limit of \( f(x) \) as \( x \) approaches 0. This is expressed formally as \( \lim_{x \to 0} \frac{1}{x} \neq L \). **Preliminary Work (3.1.4):** - Write down and explore your thought process. - Analyze how \( f(x) \) behaves as \( x \) approaches 0 from both the positive and negative sides. - Use the \( \epsilon\)–\( \delta \) definition of a limit to support your explanation. **Formal Proof (3.1.5):** Prepare a structured and rigorous mathematical proof to demonstrate the claim.
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