Short Answer

Evaluate the following limits, or explain why the limit does not exist.
$\lim_{(x, y) \to (0, 0)} \frac{x^{3} + y^{3}}{x^{2} + y^{2}}$

$\lim_{(x, y) \to (0, 0)} \frac{x^{3} + y^{3}}{x^{2} + y^{2}} = 0$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information is:

$\lim_{(x, y) \to (0, 0)} \frac{x^{3} + y^{3}}{x^{2} + y^{2}}$

Step 2. Evaluating Limits

$Since, (x, y) \to (0, 0) x^{3} + y^{3} = 0 and x^{2} + y^{2} = 0 So to find \lim_{(x, y) \to (0, 0)} \frac{x^{3} + y^{3}}{x^{2} + y^{2}}, take x = r cosθ and y = r sinθ then, (x, y) \to (0, 0) when r \to 0 Now, \frac{x^{3} + y^{3}}{x^{2} + y^{2}} = \frac{r^{3} \cos^{3} θ + r^{3} \sin^{3} θ}{r^{2} \cos^{2} θ + r^{2} \sin^{2} θ} = r (\cos^{3} θ + \sin^{3} θ) Therefore, \lim_{(x, y) \to (0, 0)} \frac{x^{3} + y^{3}}{x^{2} + y^{2}} = \lim_{r \to 0} [r (\cos^{3} θ + \sin^{3} θ)] = 0 Hence, \lim_{(x, y) \to (0, 0)} \frac{x^{3} + y^{3}}{x^{2} + y^{2}} = 0$

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Calculus

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Short Answer

Evaluate the following limits, or explain why the limit does not exist.
$\lim_{(x, y) \to (0, 0)} \frac{x^{3} + y^{3}}{x^{2} + y^{2}}$

Step by Step Solution

Step 1. Given information is:

Step 2. Evaluating Limits

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Step by Step Solution

Step 1. Given information is:

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Evaluate the following limits, or explain why the limit does not exist.
$\lim_{(x, y) \to (0, 0)} \frac{x^{3} + y^{3}}{x^{2} + y^{2}}$