arbitrary positive real number. Then f(x) - has a limit as x c. Evaluate %3D (3.1.1) 1 lim (Answer only.) Give the proof of your answer, using the e – 8 definition of limits. hought noth horo like Lecture 8 nage 18
arbitrary positive real number. Then f(x) - has a limit as x c. Evaluate %3D (3.1.1) 1 lim (Answer only.) Give the proof of your answer, using the e – 8 definition of limits. hought noth horo like Lecture 8 nage 18
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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![**Topic 3.1: Evaluating Limits**
**3.1.1 Problem Statement:**
Consider the function \( f(x) = \frac{1}{x} \), defined for all positive real numbers \( x > 0 \). Let \( c > 0 \) be an arbitrary positive real number. The function \( f(x) = \frac{1}{x} \) has a limit as \( x \to c \). Evaluate:
\[
\lim_{{x \to c}} \frac{1}{x} = \quad \text{(Answer only)}
\]
**Instructions:**
Provide a proof of your answer using the \(\epsilon - \delta\) definition of limits.
**3.1.2 Preliminary Work:**
Write your thought process here, referencing Lecture 8, page 18, to develop your understanding.
**3.1.3 Formal Proof:**
Develop a complete formal proof based on the preliminary work and the limit definition provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a8c2ebd-8c0c-4149-bd0d-58e45b4516b3%2F6501c45d-3f82-4b0d-9a63-beb3538997b9%2F4zu9d39s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic 3.1: Evaluating Limits**
**3.1.1 Problem Statement:**
Consider the function \( f(x) = \frac{1}{x} \), defined for all positive real numbers \( x > 0 \). Let \( c > 0 \) be an arbitrary positive real number. The function \( f(x) = \frac{1}{x} \) has a limit as \( x \to c \). Evaluate:
\[
\lim_{{x \to c}} \frac{1}{x} = \quad \text{(Answer only)}
\]
**Instructions:**
Provide a proof of your answer using the \(\epsilon - \delta\) definition of limits.
**3.1.2 Preliminary Work:**
Write your thought process here, referencing Lecture 8, page 18, to develop your understanding.
**3.1.3 Formal Proof:**
Develop a complete formal proof based on the preliminary work and the limit definition provided.
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