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Show that the followiny limit clees not exist lim

Show that the followiny limit clees not exist lim

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...
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**Problem Statement:** Show that the following limit does not exist: \[ \lim_{(x,y) \to (0,0)} \frac{x^2y}{x^4 + y^2} \] **Explanation:** - You are asked to demonstrate that the limit of the given function as \((x, y)\) approaches \((0, 0)\) does not exist. - The expression \(\frac{x^2y}{x^4 + y^2}\) involves both \(x\) and \(y\) in its numerator and denominator, which suggests that examining different paths towards the origin could lead to different limit values, indicating the non-existence of the overall limit.
Transcribed Image Text:**Problem Statement:** Show that the following limit does not exist: \[ \lim_{(x,y) \to (0,0)} \frac{x^2y}{x^4 + y^2} \] **Explanation:** - You are asked to demonstrate that the limit of the given function as \((x, y)\) approaches \((0, 0)\) does not exist. - The expression \(\frac{x^2y}{x^4 + y^2}\) involves both \(x\) and \(y\) in its numerator and denominator, which suggests that examining different paths towards the origin could lead to different limit values, indicating the non-existence of the overall limit.
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In this question , we can choose path y=mx and we can see that for different different value of m , limit is different., 

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