Problem 15 - Calculus with Concepts in Calculus

Question

Evaluate the limit. $$ \lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x^{3}+y^{3}+z^{3}}{x^{2}+y^{2}+z^{2}} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Observe the Form

We are given a limit expression, \( \lim_{(x, y, z) \rightarrow (0, 0, 0)} \frac{x^3 + y^3 + z^3}{x^2 + y^2 + z^2} \). This expression can potentially simplify to an indeterminate form \( \frac{0}{0} \), suggesting that a direct substitution would not work.

Step 2: Substitute along the x-axis

Evaluate the limit along the x-axis, where \( y = 0 \) and \( z = 0 \). The expression becomes \( \frac{x^3}{x^2} = x \). As \( x \to 0 \), this limit approaches 0.

Step 3: Substitute along the y-axis

Evaluate the limit along the y-axis, where \( x = 0 \) and \( z = 0 \). The expression simplifies to \( \frac{y^3}{y^2} = y \). As \( y \to 0 \), this limit also approaches 0.

Step 4: Substitute along the z-axis

Evaluate the limit along the z-axis, where \( x = 0 \) and \( y = 0 \). The expression becomes \( \frac{z^3}{z^2} = z \). As \( z \to 0 \), this limit again approaches 0.

Step 5: Substitute along the line x = y = z

Evaluate the limit along the line \( x = y = z \). Substituting, we get \( \frac{3x^3}{3x^2} = x \). As \( x \to 0 \), this limit approaches 0.

Step 6: Apply Squeeze Theorem

Consider that \(|x^3 + y^3 + z^3| \leq |x|^3 + |y|^3 + |z|^3\) and \(x^2 + y^2 + z^2 > 0\) for \((x, y, z) eq (0,0,0)\). Thus, \(\left|\frac{x^3 + y^3 + z^3}{x^2 + y^2 + z^2}\right| \leq \sqrt{x^2 + y^2 + z^2}\). As \((x, y, z) \to (0, 0, 0)\), \(\sqrt{x^2 + y^2 + z^2} \to 0\), confirming the limit is 0.

Key Concepts

Squeeze Theorem

The Squeeze Theorem is a very handy tool when dealing with limits, especially when the standard approaches fall short. To understand it, imagine that you want to evaluate a complex limit. You're not sure exactly what the function's behavior is as it approaches the desired point, but you notice it is "squeezed" between two simpler bound functions whose limits are easier to evaluate. The key idea is:

• If a function \( f(x) \) is between two other functions, \( g(x) \) and \( h(x) \), such that \( g(x) \leq f(x) \leq h(x) \) for all \( x \) in a neighborhood around a certain point (except possibly the point itself),
• And if \( \lim_{x \to a} g(x) = L \) and \( \lim_{x \to a} h(x) = L \),
• Then \( \lim_{x \to a} f(x) = L \).

In our problem, the expression \( \frac{x^3 + y^3 + z^3}{x^2 + y^2 + z^2} \) is squeezed by a simple function \( \sqrt{x^2 + y^2 + z^2} \), whose limit we know is 0 as \( (x, y, z) \to (0, 0, 0) \). Hence, the Squeeze Theorem helps us conclude the limit is 0.

Indeterminate Forms

Indeterminate forms are common in limit problems, particularly in multivariable calculus. As the term suggests, these forms result in uncertainties when we directly substitute values. The most common indeterminate forms are \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \cdot \infty \), and more. Each form indicates a limit problem where substitute operations fail to produce a clear limit.When you encountered the limit expression \( \frac{x^3 + y^3 + z^3}{x^2 + y^2 + z^2} \) with input approaching \( (0, 0, 0) \), it simplifies to \( \frac{0}{0} \), an indeterminate form. In such cases, alternative methods like the Squeeze Theorem or algebraic simplification are used to evaluate the true limit. The step-by-step path evaluations in our solution helped us see through the indeterminacy, ensuring that the limit approaches 0 in each selected path.

Pathwise Limits

Pathwise limits are an essential concept in multivariable calculus limits. They involve evaluating a limit along different paths towards a common endpoint to understand or confirm if a function has a particular limit.Consider you need to evaluate a limit at a point in a multidimensional space. Unlike single-variable limits, multivariable limits can be approached from many directions or paths. For any such limit to exist, it must have the same value along any path taken to approach the point. In our scenario, we checked the limit along multiple paths by setting all variables equal to, or some variables to be zero. The paths were:

• Along the x-axis, \( y = 0, z = 0 \)
• Along the y-axis, \( x = 0, z = 0 \)
• Along the z-axis, \( x = 0, y = 0 \)
• Along the line \( x = y = z \)

All these paths consistently approached the same limit, confirming the true multivariable limit as 0. Therefore, pathwise evaluation ensures that a consistent value is arrived at no matter which direction the point is approached from—critical when analyzing multi-directional limits.