Question

In Exercises \(19-24,\) find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x y+y z^{2}+x z^{2}}{x^{2}+y^{2}+z^{2}}\)

Short answer

The limit of the function as \((x, y, z)\) approaches \((0,0,0)\) is 0.

Step-by-step solution

Substitute the variables x, y and z

Let's consider \(x = r\cos{\theta_1}, y = r\cos{\theta_2}, z = r\cos{\theta_3}\). The r in this step is a radius which can approach zero and \(\theta_1, \theta_2, \theta_3\) are angles. This can potentially simplify the function as well as the denominator as it becomes \(r^2\). Substitution gives us the following function: \(\lim _{r \rightarrow 0} \frac{r^3(\cos{\theta_1}\cos{\theta_2}+\cos{\theta_2}\cos^2{\theta_3}+\cos{\theta_1}\cos^2{\theta_3})}{r^2}\)

Carry out limit calculation with r

We can cancel out the r squared in the numerator and denominator. The function then becomes \(\lim _{r \rightarrow 0} r(\cos{\theta_1}\cos{\theta_2}+\cos{\theta_2}\cos^2{\theta_3}+\cos{\theta_1}\cos^2{\theta_3})\). Taking the limit as r approaches 0, the entire function will approach 0.

Interpret final result

Since our approach resulted in a concrete number (in this case, 0), we can say that the limit of the function as it approaches (0,0,0) is 0.

Key concepts

Limit Calculation

Understanding the calculation of multivariable limits involves assessing how a function behaves as the input variables approach specific values from all possible directions. In the given exercise, we are tasked with calculating the limit of a three-dimensional function as the point \(x, y, z\) gets infinitesimally close to the origin \(0,0,0\).

The original function involves a ratio of two expressions, and directly substituting the point \(0,0,0\) into the function would yield an indeterminate form of \(\frac{0}{0}\). Spotting indeterminate forms is essential in limit calculation as it indicates the need for more advanced methods to evaluate the limit, such as algebraic manipulation or coordinate transformation. In this case, simplifying the limit calculation includes changing the variables to a more advantageous coordinate system, facilitating the process of finding the limit—if it exists.

Spherical Coordinates Substitution

Transforming Cartesian coordinates \( (x, y, z) \) into spherical coordinates \( (r, \theta_1, \theta_2, \theta_3) \) is a commonly used technique to simplify the evaluation of limits for multivariable functions that involve the distance from the origin.

The substitution involves setting \(x = r\cos{\theta_1}\), \(y = r\cos{\theta_2}\), and \(z = r\cos{\theta_3}\), where \(r\) signifies the distance from the origin to the point in space, and \(\theta\)'s represent angles relative to the axes. This simplifies the original expression significantly because the terms in the denominator transform to \(r^2\), reducing the complexity of the expression. When substituting back into the limit, it's often the case that an \(r\) will factor out and cancel, as seen in the solution. This manipulation brings us closer to determining whether the limit exists and, if so, what its value is.

Limit Existence

The question of a limit's existence is crucial when dealing with functions that approach a certain point. For a limit to exist at a particular point, the function must approach the same value from all possible paths leading to that point.

In our exercise, after substituting spherical coordinates and simplifying, we were left with a function multiplied by \(r\) which approaches zero as \(r\) does. Regardless of the values of the angles \(\theta_1, \theta_2, \theta_3\), the remaining factor of \(r\) will lead the entire function towards zero. Thus, we conclude that as the point \( (x, y, z) \) gets infinitely close to \( (0,0,0) \) from any direction, the function itself tends towards zero. This confirms the existence of the limit, which is a fundamental aspect of continuity and crucial for ensuring that various mathematical operations can be performed on the function, such as differentiation and integration. Identifying and confirming the existence of a limit provides both a sense of closure and a gateway to future analysis on the function.