Using Polar Coordinates To Find Limits

Using Polar Coordinates to Find Limits

Finding limits of multivariable functions can sometimes be tricky, especially when approaching the origin along different paths. This article explores a powerful technique: using polar coordinates to evaluate limits, particularly those involving indeterminate forms at the origin. This method helps circumvent the complexities of approaching (0,0) along various paths, providing a more streamlined and often decisive approach to limit evaluation.

Understanding the Challenge: Different Paths, Different Limits

When evaluating the limit of a multivariable function f(x, y) as (x, y) approaches (0, 0), simply substituting x = 0 and y = 0 isn't always sufficient. The limit might exist only if the function approaches the same value regardless of the path taken to (0, 0). Different paths can yield different limits, signifying the limit does not exist. This makes direct substitution unreliable.

The Polar Coordinate Transformation: A Smoother Path

Polar coordinates provide an elegant solution. We transform the Cartesian coordinates (x, y) into polar coordinates (r, θ) using the transformations:

• x = r cos θ
• y = r sin θ

Where r represents the distance from the origin and θ represents the angle. Crucially, as (x, y) approaches (0, 0), r approaches 0, regardless of the angle θ.

Applying Polar Coordinates to Evaluate Limits

The process involves these steps:

1. Convert to Polar Coordinates: Substitute x = r cos θ and y = r sin θ into the function f(x, y).

2. Simplify: Simplify the resulting expression. The goal is to express the function solely in terms of r. Often, terms involving θ will cancel out or become insignificant as r approaches 0.

3. Evaluate the Limit: Take the limit as r approaches 0. If the limit exists and is independent of θ, then this is the limit of the original function as (x, y) approaches (0, 0). If the limit depends on θ or does not exist, then the original limit does not exist.

Examples: Illuminating the Technique

Let's illustrate with some examples:

Example 1: A Limit that Exists

Consider the limit:

lim (x,y)→(0,0) (x²y) / (x⁴ + y²)

1. Convert to Polar Coordinates:

   (r²cos²θ * rsinθ) / (r⁴cos⁴θ + r²sin²θ)

2. Simplify:

   (r³cos²θsinθ) / (r²(r²cos⁴θ + sin²θ)) = (r cos²θsinθ) / (r²cos⁴θ + sin²θ)

3. Evaluate the Limit:

   As r approaches 0, the numerator approaches 0, while the denominator approaches sin²θ (unless sin θ = 0, which is a specific path). Therefore, the limit is 0.

Example 2: A Limit that Does Not Exist

Consider the limit:

lim (x,y)→(0,0) x / (x + y)

1. Convert to Polar Coordinates:

   (rcosθ) / (rcosθ + rsinθ)

2. Simplify:

   cosθ / (cosθ + sinθ)

3. Evaluate the Limit:

   The limit depends on θ. For example, if θ = 0, the limit is 1; if θ = π/2, the limit is 0. Since the limit depends on the path (the value of θ), the limit does not exist.

Conclusion: A Powerful Tool in Your Limit-Finding Arsenal

Polar coordinates offer a powerful and often necessary technique for evaluating limits of multivariable functions, particularly those involving indeterminate forms at the origin. By transforming to polar coordinates, we can often simplify complex expressions and determine whether a limit exists and, if so, its value. This method helps overcome the challenges posed by considering multiple paths to the origin, offering a more systematic and efficient approach to limit evaluation. Mastering this technique is essential for a deeper understanding of multivariable calculus and its applications.