Q 44. In Exercises 41鈥�46, use polar ... [FREE SOLUTION]

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Chapter 12: Q 44. (page 931)

In Exercises 41鈥�46, use polar coordinates to analyze the given limits.
$\lim_{(x, y) 鈫� (0, 0)} \frac{x^{2} y^{3}}{x^{4} + 2 x^{2} y^{2} + y^{4}}$

Short Answer

Expert verified

The value of $\lim_{(x, y) 鈫� (0, 0)} \frac{x^{2} y^{3}}{x^{4} + 2 x^{2} y^{2} + y^{4}}$ is 0.

Step by step solution

Step 1. Given information.

We have given expression : $\lim_{(x, y) 鈫� (0, 0)} \frac{x^{2} y^{3}}{x^{4} + 2 x^{2} y^{2} + y^{4}}$

Step 2. Use polar coordinates to analyze the given limits.

The relation between the rectangular coordinates $(x, y)$ and the polar coordinates $(r, 胃)$ is

$x = r \cos 胃 y = r \sin 胃$

On substituting values of $x$ and $y$ we get.

$\lim_{(x, y) 鈫� (0, 0)} \frac{x^{2} y^{3}}{x^{4} + 2 x^{2} y^{2} + y^{4}} = \lim_{r 鈫� 0} \frac{r^{2} \cos^{2} 胃 (r^{3} \sin^{3} 胃)}{r^{4} \cos^{4} 胃 + 2 r^{2} \cos^{2} 胃 (r^{2} \sin^{2} 胃) + r^{4} \sin^{4} 胃} = \lim_{r 鈫� 0} \frac{r^{5} \cos^{2} 胃 \sin^{3} 胃}{r^{4} (\cos^{4} 胃 + \sin^{4} 胃 + 2 \sin^{2} 胃 \cos^{2} 胃)} = \lim_{r 鈫� 0} \frac{r^{} \cos^{2} 胃 \sin^{3} 胃}{(\cos^{4} 胃 + \sin^{4} 胃 + 2 \sin^{2} 胃 \cos^{2} 胃)} = 0$

Since the value of $\lim_{(x, y) 鈫� (0, 0)} \frac{x^{2} y^{3}}{x^{4} + 2 x^{2} y^{2} + y^{4}}$ is $0$ .

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91影视

In Exercises 41鈥�46, use polar coordinates to analyze the given limits.
$\lim_{(x, y) 鈫� (0, 0)} \frac{x^{2} y^{3}}{x^{4} + 2 x^{2} y^{2} + y^{4}}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use polar coordinates to analyze the given limits.

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91影视

In Exercises 41鈥�46, use polar coordinates to analyze the given limits.lim(x,y)鈫�0,0x2y3x4+2x2y2+y4

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use polar coordinates to analyze the given limits.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Logic and Functions

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

In Exercises 41鈥�46, use polar coordinates to analyze the given limits.
$\lim_{(x, y) 鈫� (0, 0)} \frac{x^{2} y^{3}}{x^{4} + 2 x^{2} y^{2} + y^{4}}$