Chapter 12: Q 13. (page 930)

Show that when C is either the $x - o r y - a x i s$ , we have $\lim_{\underset{C_{}}{(x, y) \to (0, 0)}} \frac{x^{2} y}{x^{4} + y^{2}} = 0$ .

Short Answer

Expert verified

It can be shown that limit is zero at both $x = 0 a n d y = 0$ .

Step by step solution

Given Information

Consider that the limit is $\lim_{\underset{C_{}}{(x, y) \to (0, 0)}} \frac{x^{2} y}{x^{4} + y^{2}} = 0$

The objective is to prove that the limiting value is

0

, when C is either the role="math" localid="1653469066356"

x - o r y - a x i s

Evaluating the limit 1

Assume that curve C is on the $x - a x i s$ , with $y = 0$ . Calculate the function limit as you approach point $(0, 0)$ on the curve C as follows:

$\lim_{\underset{C_{}}{(x, y) \to (0, 0)}} \frac{x^{2} y}{x^{4} + y^{2}} = \lim_{\underset{}{(x) \to (0)}} \frac{x^{2} (0)}{x^{4} + {(0)}^{2}} = \lim_{\underset{}{(x) \to (0)}} \frac{0}{x^{4}} = 0$

Evaluating the limit 2

Assume that curve C is on the $y - a x i s$ , with $x = 0$ . Calculate the function limit as you approach point $(0, 0)$ on the curve C as follows:

$\lim_{\underset{C_{}}{(x, y) \to (0, 0)}} \frac{x^{2} y}{x^{4} + y^{2}} = \lim_{\underset{}{(y) \to (0)}} \frac{{(0)}^{2} y}{{(0)}^{4} + y^{2}} = \underset{y = 0}{l i m} \frac{0}{y^{2}} = 0$

Hence, the limiting value of the function $\frac{x^{2} y}{x^{4} + y^{2}}$ when $(x, y) \to (0, 0)$ along the curve C, here, C is either the $x - o r y - a x i s$ is $0$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Show that when C is either the $x - o r y - a x i s$ , we have $\lim_{\underset{C_{}}{(x, y) \to (0, 0)}} \frac{x^{2} y}{x^{4} + y^{2}} = 0$ .

Short Answer

Step by step solution

Given Information

Evaluating the limit 1

Evaluating the limit 2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Geometry

Discrete Mathematics

Logic and Functions

Calculus

Study anywhere. Anytime. Across all devices.

Show that when C is either the x-ory-axis, we havelim(x,y)→(0,0)Cx2yx4+y2=0.

Short Answer

Step by step solution

Given Information

Evaluating the limit 1

Evaluating the limit 2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Geometry

Discrete Mathematics

Logic and Functions

Calculus

Study anywhere. Anytime. Across all devices.

Show that when C is either the $x - o r y - a x i s$ , we have $\lim_{\underset{C_{}}{(x, y) \to (0, 0)}} \frac{x^{2} y}{x^{4} + y^{2}} = 0$ .