Chapter 12: Q. 10 (page 989)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information is:

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

Step 2. Evaluating Limits

$Since, (x, y, z) \to (0, 0, 0) x^{2} + y^{2} \to 0 and x^{2} + y^{2} + z^{2} \to 0 So to find \lim_{(x, y, z) \to (0, 0, 0)} \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}}, take x = r cosθ sinϕ, y = r sinθ sinϕ and z = r cosϕ then, (x, y, z) \to (0, 0, 0) when r \to 0 Now, \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}} = \frac{r^{2} \cos^{2} θ \sin^{2} ϕ + r^{2} \sin^{2} θ \sin^{2} ϕ}{r^{2} \cos^{2} θ \sin^{2} ϕ + r^{2} \sin^{2} θ \sin^{2} ϕ + r^{2} \cos^{2} ϕ} \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}} = \frac{\sin^{2} ϕ}{\sin^{2} ϕ + \cos^{2} ϕ} = \sin^{2} ϕ Therefore, \lim_{(x, y, z) \to (0, 0, 0)} \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}} = \sin^{2} ϕ, which depends on ϕ For different values of ϕ, there are different values of \sin^{2} ϕ Hence, \lim_{(x, y, z) \to (0, 0, 0)} \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}} = does not exist$

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

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)→(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

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Evaluate the following limits, or explain why the limit does not exist.
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