Problem 61 - Calculus Volume 3

Question

For the following exercises, find the limit of the function. $$\lim _{(x, y) \rightarrow(1,2)} \frac{5 x^{2} y}{x^{2}+y^{2}}$$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Identify the Limit Statement

We need to find the limit \( \lim_{(x, y) \to (1, 2)} \frac{5x^2 y}{x^2 + y^2} \). This expression involves two variables, so we'll approach it using paths or direct substitution if possible.

Step 2: Evaluate Using Direct Substitution

First, attempt direct substitution by plugging \( x = 1 \) and \( y = 2 \) into the function. This yields: \( \frac{5(1)^2(2)}{(1)^2 + (2)^2} = \frac{5 \cdot 1 \cdot 2}{1 + 4} \).

Step 3: Simplify the Expression

Simplify the expression from the previous step: \( \frac{10}{5} = 2 \). This confirms the limit arrives at a defined number by direct substitution.

Key Concepts

Direct Substitution

Direct substitution is one of the simplest methods for finding limits. It involves plugging the values of the approaching variables directly into the function to see if it gives a finite limit. It works best when the function is continuous at the point of interest.

In the given exercise, we have a two-variable function, \( \frac{5x^2 y}{x^2 + y^2} \), and the task is to find the limit as \((x, y)\) approaches \((1, 2)\). To apply direct substitution, we simply substitute \(x = 1\) and \(y = 2\) into the expression:

\[\frac{5 \times 1^2 \times 2}{1^2 + 2^2} = \frac{10}{5} = 2.\]This substitution process confirms the limit is 2, which means the function at the point \((1, 2)\) tends towards this value. Direct substitution is straightforward, but it doesn't always work, especially if the function is undefined at the point of interest.

Two-Variable Functions

Two-variable functions involve equations with both \(x\) and \(y\) as variables. These types of functions are typically represented as surfaces or planes in three-dimensional space.

The function in our exercise, \( \frac{5x^2 y}{x^2 + y^2} \), is a classic example of a two-variable function. Here, the numerator represents a scaled form of the area dependent on \(x^2 y\), while the denominator represents a combination of squared terms \(x^2 + y^2\).

When working with limits of two-variable functions, keep in mind:

• You're essentially analyzing the behavior of a surface as it nears a specific point in the \((x, y)\) plane.
• The limit can depend on the path taken by \((x, y)\) to approach the point, which complicates matters over single-variable functions.
• Ensuring understanding of surfaces and their corresponding equation structures can help visualize how substitutions and paths affect the limit.

Such functions require more attention compared to single-variable calculus because their limits may not exist if different paths yield different results.

Path Approach

When direct substitution fails or a function likely has different behaviors from various directions, the path approach is a viable option. This method involves checking the limit by approaching the target point from different paths. If the results vary, the overall limit does not exist.

In the context of our exercise, if direct substitution hadn't been successful or provided indeterminate results, we could use the path approach to further investigate. Here’s how it works:

• Straight line paths: Choose lines like \(y = mx + b\) that lead to the point \((1, 2)\) and substitute these into the function to see if they agree on a limit.
• Curved paths: Functions like \(y = x^2\) or \(x = \sin(y)\) can be used as alternative paths to examine behavior.
• Comparative analysis: Cross-checking results from multiple paths is crucial to conclude if a consistent limit exists.

If every reasonable path to the point \((1, 2)\) provides the same result of 2 (as found by our direct substitution), the limit can be confidently stated to be 2. Path approach verifies the stability of a limit across multidimensional paths.