Chapter 12: Q. 34 (page 931)

Evaluate the limits in Exercises 33–40 if they exist.
$\lim_{(x, y, z) \to (3, - 4, π / 4)} \frac{x^{2} y^{}}{\tan z}$

Short Answer

Expert verified

The limit is $- 36$ .

Step by step solution

Given Information

Consider the function $\lim_{(x, y, z) \to (3, - 4, π / 4)} \frac{x^{2} y^{}}{\tan z}$

The goal is to assess $\lim_{(x, y, z) \to (3, - 4, π / 4)} \frac{x^{2} y^{}}{\tan z}$ $f$ if it exists.

Defining the limit

Consider the following assertion:

Consider a three-variable function $f (x, y, z)$ that is continuous at all points on $R^{3}$ . The limit of the function $f (x, y, z)$ as $(x, y, z) \to (x_{0}, y_{0}, z_{0})$ is therefore defined as

$\lim_{(x, y, z) \to (x_{0}, y_{0}, z_{0})} f (x, y, z) = f (x_{0}, y_{0}, z_{0})$

Evaluating the limit

Because $x^{2} y$ is a two-variable polynomial function, it is continuous for every point on $R^{3}$ , and the transcendental number $\tan z$ is also continuous for every point on $R^{3}$ .

Thus, where $\frac{x^{2} y}{\tan z}$ is continuous, where the rational function $\frac{x^{2} y}{\tan z}$ is defined .

As a result of the statement,

$\lim_{(x, y, z) \to (3, - 4, π / 4)} \frac{x^{2} y^{}}{\tan z} = \frac{{(3)}^{2} (- 4)}{\tan π / 4} = - 36$

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Evaluate the limits in Exercises 33–40 if they exist.
$\lim_{(x, y, z) \to (3, - 4, π / 4)} \frac{x^{2} y^{}}{\tan z}$

Short Answer

Step by step solution

Given Information

Defining the limit

Evaluating the limit

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Evaluate the limits in Exercises 33–40 if they exist.lim(x,y,z)→(3,-4,π/4)x2ytanz

Short Answer

Step by step solution

Given Information

Defining the limit

Evaluating the limit

One App. One Place for Learning.

Most popular questions from this chapter

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Probability and Statistics

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Theoretical and Mathematical Physics

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Evaluate the limits in Exercises 33–40 if they exist.
$\lim_{(x, y, z) \to (3, - 4, π / 4)} \frac{x^{2} y^{}}{\tan z}$