Q. 34 - Calculus

Question

Evaluate the limits in Exercises 33–40 if they exist. 

$\underset{(x,y,z)\to (3,-4,\pi /4)}{\lim }\frac{x^2{y}^{}}{\tan z}$

Step-by-Step Solution

Answer

The limit is $-36$.

Step 1: Given Information

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

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

Step 2: 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

$\underset{(x,y,z)\to (x_0,y_0,z_0)}{\lim }f(x,y,z)=f(x_0,y_0,z_0)$

Step 3: 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,

$\underset{(x,y,z)\to (3,-4,\pi /4)}{\lim }\frac{x^2{y}^{}}{\tan z}=\frac{{\left(3\right)}^2(-4)}{\tan \pi /4}=-36$