Problem 18 - Calculus with Concepts in Calculus

Question

Determine whether \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} \sin x}{x^{2}+y^{2}}\) exists.

Step-by-Step Solution

Answer

The limit exists and is equal to 0.

Step 1: Understand the Problem

We need to determine if the limit exists as \((x, y)\) approaches \((0,0)\) for the function \(f(x, y) = \frac{x^{2} \sin x}{x^{2}+y^{2}}\). A limit exists at a point if it approaches the same value regardless of the path taken towards that point.

Step 2: Choose Paths to Approach the Limit

First, we'll test a path by letting \(y = 0\). The limit becomes \(\lim _{x \to 0} \frac{x^{2} \sin x}{x^{2}}\), which simplifies to \(\lim _{x \to 0} \sin x = 0\), as \(\sin x\) is continuous at 0.

Step 3: Test Another Path

Now, consider the path \(x = 0\). This gives us \(\lim _{y \to 0} \frac{0 \cdot \sin(0)}{y^2} = 0\). Again, the limit is 0 along this path.

Step 4: Test the Polar Coordinates Path

Convert to polar coordinates: set \(x = r \cos \theta\) and \(y = r \sin \theta\). The expression becomes \(\frac{(r \cos \theta)^{2} \sin(r \cos \theta)}{r^2}\), which simplifies to \(r\cos^2\theta \cdot \frac{\sin(r \cos \theta)}{r \cos \theta}\). As \(r \to 0\), \(\frac{\sin(r \cos \theta)}{r \cos \theta} \to 1\), which shows the expression approaches 0 because \(r \to 0\) implies \(r\cos^2\theta \to 0\).

Step 5: Conclusion

Since every path tested results in the limit being 0, we conclude \(\lim _{(x, y) \to (0,0)} \frac{x^{2} \sin x}{x^{2}+y^{2}} = 0\). The limit exists and is 0.

Key Concepts

Limits in Multivariable Calculus

In multivariable calculus, limits help us understand the behavior of functions with more than one variable as they approach a particular point. Unlike single variable calculus where limits involve just one variable, here, we have two or more.
To determine if a limit exists for a function of two variables, such as \(f(x,y)\), we need to consider all possible paths approaching a target point, like \((0,0)\). A limit exists if, and only if, the function approaches the same value, regardless of the path taken towards the point.
This concept is crucial because, in multivariable settings, functions can behave differently along different approaches. If different paths give different results when calculating the limit, then it does not exist. Test multiple paths to ensure consistency.

Polar Coordinates

Polar coordinates are an alternative to the Cartesian coordinate system, providing a useful framework especially for circular or radial symmetry. Instead of using \((x, y)\), polar coordinates use \((r, \theta)\), where \(r\) represents the radial distance from the origin and \(\theta\) denotes the angle from the positive x-axis.
This conversion is achieved through the transformations:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

By using polar coordinates, we can often simplify complex multivariable limit problems, especially those involving symmetry around the origin.
In our example, converting to polar coordinates helped streamline the limit calculation, confirming the consistency of results with other paths like \(y = 0\) and \(x = 0\). As \(r\) approaches zero, the function simplifies, revealing that the overall limit converges to zero.

Path Independence in Limits

Path independence is an essential concept when determining multivariable limits. If a limit has path independence, it means the function approaches the same value across all possible paths towards a specific point. This is a crucial factor in proving the existence of a limit.
In scenarios where path dependence exists, different paths yield differing results, indicating the non-existence of a multivariable limit.
To test path independence, consider substituting values or converting coordinates, like using polar coordinates, to evaluate different approaches to the point.
For example, finding a limit along the x-axis, y-axis, and through polar coordinates may help illustrate path independence. If all these approaches generate the same limit, as they did in our exercise, it confirms the true existence of the limit.