Q 13. Show that when C is either the x... [FREE SOLUTION]

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Chapter 12: Q 13. (page 930)

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

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

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:

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

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:

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

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

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91影视

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

Short Answer

Step by step solution

Given Information

Evaluating the limit 1

Evaluating the limit 2

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Show that when C is either the x-ory-axis, we havelim(x,y)鈫�(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

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Theoretical and Mathematical Physics

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Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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