Q. 10 Evaluate the following limits, o... [FREE SOLUTION]

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Chapter 12: Q. 10 (page 989)

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

Short Answer

Expert verified

$\lim_{(x, y, z) 鈫� (0, 0, 0)} \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}} = does not exists$

Step by step solution

Step 1. Given information is:

$\lim_{(x, y, z) 鈫� (0, 0, 0)} \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}}$

Step 2. Evaluating Limits

$Since, (x, y, z) 鈫� (0, 0, 0) x^{2} + y^{2} 鈫� 0 and x^{2} + y^{2} + z^{2} 鈫� 0 So to find \lim_{(x, y, z) 鈫� (0, 0, 0)} \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}}, take x = r 肠辞蝉胃蝉颈苍蠒, y = r 蝉颈苍胃蝉颈苍蠒 and z = r 肠辞蝉蠒 then, (x, y, z) 鈫� (0, 0, 0) when r 鈫� 0 Now, \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}} = \frac{r^{2} \cos^{2} 胃 \sin^{2} 蠒 + r^{2} \sin^{2} 胃 \sin^{2} 蠒}{r^{2} \cos^{2} 胃 \sin^{2} 蠒 + r^{2} \sin^{2} 胃 \sin^{2} 蠒 + r^{2} \cos^{2} 蠒} \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}} = \frac{\sin^{2} 蠒}{\sin^{2} 蠒 + \cos^{2} 蠒} = \sin^{2} 蠒 Therefore, \lim_{(x, y, z) 鈫� (0, 0, 0)} \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}} = \sin^{2} 蠒, which depends on 蠒 For different values of 蠒, there are different values of \sin^{2} 蠒 Hence, \lim_{(x, y, z) 鈫� (0, 0, 0)} \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}} = does not exist$

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91影视

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

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Evaluating Limits

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Evaluate the following limits, or explain why the limit does not exist.lim(x,y,z)鈫�(0,0,0)x2+y2x2+y2+z2

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Evaluating Limits

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Calculus

Statistics

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.

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