Answered: Find the limit (if it exists) or show…

Math

Calculus

Find the limit (if it exists) or show that the limit does not exist lim(x,y)-->(0,0) x2+y2/x2-6y2

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.3: Algebraic Expressions

Problem 20E

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Concept explainers

Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as:

Question

Find the limit (if it exists) or show that the limit does not exist

lim_{(x,y)-->(0,0)} x²+y²/x²-6y²

Expert Solution

Step 1

Here, we will show that the limit of $\lim_{(x, y) \to (0, 0)} (\frac{x^{2} + y^{2}}{x^{2} - 6 y^{2}})$ doesn't exists.

Along x=0:

$\lim_{(x, y) \to (0, 0)} (\frac{x^{2} + y^{2}}{x^{2} - 6 y^{2}}) F o r x = 0, = \lim_{y \to 0} \frac{0 + y^{2}}{0 - 6 y^{2}} = - \frac{1}{6}$

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