Homework Help is Here – Start Your Trial Now!
Math
Calculus
Use the E-8 definition of a limit to prove lim Xy (Xy) →(0,0) x² +y²

Use the E-8 definition of a limit to prove lim Xy (Xy) →(0,0) x² +y²

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
See similar textbooks
Question
**Topic: Proving Limits Using the Epsilon-Delta Definition** Use the \(\varepsilon - \delta\) definition of a limit to prove: \[ \lim_{(x,y) \to (0,0)} \frac{xy}{\sqrt{x^2 + y^2}} = 0 \] **Explanation:** This statement asks to prove a two-variable limit using the epsilon-delta (\(\varepsilon - \delta\)) definition. The expression \(\frac{xy}{\sqrt{x^2 + y^2}}\) must approach 0 as the point \((x, y)\) approaches \((0, 0)\). In the context of two variables, we say the limit of \(f(x, y)\) as \((x, y) \to (0, 0)\) is \(L\) if for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < \sqrt{x^2 + y^2} < \delta\), then \(|f(x, y) - L| < \varepsilon\). Here, you need to demonstrate that: For any \(\varepsilon > 0\), a \(\delta > 0\) exists such that if \(0 < \sqrt{x^2 + y^2} < \delta\), then: \[ \left| \frac{xy}{\sqrt{x^2 + y^2}} - 0 \right| < \varepsilon \] This involves manipulating the inequality and finding conditions in terms of \(\delta\) to ensure that the expression can be made arbitrarily small, hence showing it approaches zero near \((0, 0)\).
Transcribed Image Text:**Topic: Proving Limits Using the Epsilon-Delta Definition** Use the \(\varepsilon - \delta\) definition of a limit to prove: \[ \lim_{(x,y) \to (0,0)} \frac{xy}{\sqrt{x^2 + y^2}} = 0 \] **Explanation:** This statement asks to prove a two-variable limit using the epsilon-delta (\(\varepsilon - \delta\)) definition. The expression \(\frac{xy}{\sqrt{x^2 + y^2}}\) must approach 0 as the point \((x, y)\) approaches \((0, 0)\). In the context of two variables, we say the limit of \(f(x, y)\) as \((x, y) \to (0, 0)\) is \(L\) if for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < \sqrt{x^2 + y^2} < \delta\), then \(|f(x, y) - L| < \varepsilon\). Here, you need to demonstrate that: For any \(\varepsilon > 0\), a \(\delta > 0\) exists such that if \(0 < \sqrt{x^2 + y^2} < \delta\), then: \[ \left| \frac{xy}{\sqrt{x^2 + y^2}} - 0 \right| < \varepsilon \] This involves manipulating the inequality and finding conditions in terms of \(\delta\) to ensure that the expression can be made arbitrarily small, hence showing it approaches zero near \((0, 0)\).
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Limits and Continuity
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS