Problem 20 - Calculus Early Transcendentals

Question

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x^{2}+2 y^{2}+3 z^{2}}{x^{2}+y^{2}+z^{2}}$$

Step-by-Step Solution

Answer

The limit does not exist because different paths give different results.

Step 1: Introduction to the Limit

We are asked to find the limit \( \lim_{(x, y, z) \to (0,0,0)} \frac{x^2 + 2y^2 + 3z^2}{x^2 + y^2 + z^2} \). This is a multivariable limit problem where the function depends on three variables \(x, y,\) and \(z\). Our goal is to determine if the value of the limit exists and is the same regardless of the path taken to approach \((0, 0, 0)\).

Step 2: Checking Along Standard Paths

First, we consider the approach along the x-axis by letting \( y = 0 \) and \( z = 0 \). This simplifies the limit to \( \lim_{x \to 0} \frac{x^2}{x^2} = 1 \). Next, approach along the y-axis by letting \( x = 0 \) and \( z = 0 \), which gives \( \lim_{y \to 0} \frac{2y^2}{y^2} = 2 \), and then along the z-axis by letting \( x = 0 \) and \( y = 0 \), leading to \( \lim_{z \to 0} \frac{3z^2}{z^2} = 3 \).

Step 3: Conclusion From Different Paths

Since the limit depends on the path taken to approach \((0, 0, 0)\) -- yielding different results such as 1, 2, and 3 along different axes -- we conclude that the limit does not exist. A limit would exist only if the value was the same through all paths.

Key Concepts

Path-dependent limits

When dealing with multivariable limits, it is crucial to determine whether the limit is path-dependent or not. Imagine trying to reach the origin point (0, 0, 0) in three-dimensional space. There are infinitely many paths you could take to approach this point. In multivariable calculus, if the expression you are evaluating behaves differently depending on the path taken, then the limit does not exist.
For example, in our exercise, as we consider different paths to approach the origin, such as along the x-axis, y-axis, or z-axis, the resulting limit values are different: 1, 2, and 3, respectively. This shows path dependency.

• Path-independent limits: The limit is the same, regardless of the path or direction taken to approach the point.
• Path-dependent limits: Different paths result in different limit values, indicating the absence of a definite limit.

Identifying path dependency is a powerful technique in confirming that a multivariable limit does not exist.

Coordinate axes paths

One of the simplest ways to approach solving a multivariable limit problem is by utilizing the coordinate axes. This means evaluating the limit by setting two variables to zero and solving the expression along each axis individually.
In the provided exercise, we considered the x-axis by setting both y and z to zero. This gave us a simpler, single-variable limit problem: \[ \lim_{x \to 0} \frac{x^2}{x^2} = 1 \] Similarly, along the y-axis and z-axis, we set x = 0, z = 0, and x = 0, y = 0, respectively, simplifying our problem to mere single-paper computations:

• y-axis: \( \lim_{y \to 0} \frac{2y^2}{y^2} = 2 \)
• z-axis: \( \lim_{z \to 0} \frac{3z^2}{z^2} = 3 \)

Approaching via these paths reveals critical insights into whether the limit can be consistent across different directions, helping to identify if a limit doesn't exist.

Limit existence criteria

To determine if a limit exists in multivariable calculus, a fundamental criterion is that the result must be consistent across all potential paths to the point of interest—in this case, the origin. The expression should yield the same limit value, irrespective of the path taken to approach (0,0,0).
In practice:

• Calculate the limit by approaching along straightforward lines or paths, such as coordinate axes or linear paths where one variable is expressed in terms of others.
• If you obtain different results from different paths, conclude readily that the limit does not exist.

In our example, since the expression resulted in different values (1, 2, and 3) depending on which path was chosen, the criteria for a definite limit were not met. Through this strategy, you can efficiently assess whether a multivariable limit can be accepted as existent or dismissed as non-existent due to path variance.