Answered: 2ry (z.9)–{(0,0) 2² + yª Show that the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Please show the limit along x axis; y=0

Please show the limit along y axis; x=0

Please show the limit along y=Squre root of x.

Please give details as much as possible.

Show that the limit lim
2ny²
does not exist.
(z.19)-(0,0) a² + y4

Expert Solution

Step 1

The objective is to show that $\underset{(x, y) \to (0, 0)}{l i m} \frac{2 x y^{2}}{x^{2} + y^{4}}$ does not exist

The limit on x axis is,

$\begin{array}{rcl} \underset{y \to 0}{l i m} \frac{2 x y^{2}}{x^{2} + y^{4}} & = & \underset{y \to 0}{l i m} \frac{0}{x^{2} + 0} \\ = & \underset{y \to 0}{l i m} 0 \\ = & 0 \end{array}$

The limit along y axis is,

$\begin{array}{rcl} \underset{x \to 0}{l i m} \frac{2 x y^{2}}{x^{2} + y^{4}} & = & \underset{x \to 0}{l i m} \frac{0}{y^{4}} \\ = & \underset{x \to 0}{l i m} 0 \\ = & 0 \end{array}$

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