Let {z} be a sequence of complex numbers (z = x + ¡y, where x, y_€ R)and z € C. Give the 12 E- N definition of " lim z = z "and show that if lim z = z and lim w = w then 72 11 → 00 12 12 1200 12100 lim (z₁+w) = z + w 12 → 00 € - N definition of limit If XR, a is a limit point of X, and f: X → R, say that the limit of f on X at a is the number L provided that for every e > 0 there is a 8 >0 such that f(x) — L| < €whenever xe X and 0 < |ax|< 8.
Let {z} be a sequence of complex numbers (z = x + ¡y, where x, y_€ R)and z € C. Give the 12 E- N definition of " lim z = z "and show that if lim z = z and lim w = w then 72 11 → 00 12 12 1200 12100 lim (z₁+w) = z + w 12 → 00 € - N definition of limit If XR, a is a limit point of X, and f: X → R, say that the limit of f on X at a is the number L provided that for every e > 0 there is a 8 >0 such that f(x) — L| < €whenever xe X and 0 < |ax|< 8.
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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![**Complex Numbers and Limits**
Let \(\{z_n\}\) be a sequence of complex numbers \((z_n = x_n + iy_n, \text{ where } x_n, y_n \in \mathbb{R})\) and \(z \in \mathbb{C}\).
### Definition and Theorem
**ε – N Definition of Limit**
To define \(\lim_{n \to \infty} z_n = z\), show that if \(\lim_{n \to \infty} z_n = z\) and \(\lim_{n \to \infty} w_n = w\), then:
\[
\lim_{n \to \infty} (z_n + w_n) = z + w
\]
### Limit Definition Explanation
**ε – N Definition of Limit**
If \(X \subseteq \mathbb{R}\), \(a\) is a limit point of \(X\), and \(f: X \to \mathbb{R}\), say that the limit of \(f\) on \(X\) at \(a\) is the number \(L\) provided that for every \(\varepsilon > 0\) there is a \(\delta > 0\) such that \(|f(x) - L| < \varepsilon\) whenever \(x \in X\) and \(0 < |a - x| < \delta\).
### Analysis:
The text outlines the formal definitions needed to establish limits of sequences in the complex plane and explains how the ε – N definition of limits applies to functions within real numbers, adapting the core understanding to complex numbers. The relationship and operations on limits are fundamental to understanding continuous functions and sequences in both real and complex analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e8ed93-7bef-4409-89ed-52264f64a27e%2F7b87af2f-ba72-466c-a3c4-847a5c363562%2Fuv7ar58_processed.png&w=3840&q=75)
Transcribed Image Text:**Complex Numbers and Limits**
Let \(\{z_n\}\) be a sequence of complex numbers \((z_n = x_n + iy_n, \text{ where } x_n, y_n \in \mathbb{R})\) and \(z \in \mathbb{C}\).
### Definition and Theorem
**ε – N Definition of Limit**
To define \(\lim_{n \to \infty} z_n = z\), show that if \(\lim_{n \to \infty} z_n = z\) and \(\lim_{n \to \infty} w_n = w\), then:
\[
\lim_{n \to \infty} (z_n + w_n) = z + w
\]
### Limit Definition Explanation
**ε – N Definition of Limit**
If \(X \subseteq \mathbb{R}\), \(a\) is a limit point of \(X\), and \(f: X \to \mathbb{R}\), say that the limit of \(f\) on \(X\) at \(a\) is the number \(L\) provided that for every \(\varepsilon > 0\) there is a \(\delta > 0\) such that \(|f(x) - L| < \varepsilon\) whenever \(x \in X\) and \(0 < |a - x| < \delta\).
### Analysis:
The text outlines the formal definitions needed to establish limits of sequences in the complex plane and explains how the ε – N definition of limits applies to functions within real numbers, adapting the core understanding to complex numbers. The relationship and operations on limits are fundamental to understanding continuous functions and sequences in both real and complex analysis.
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