Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}}\)

Step-by-Step Solution

Answer

The limit is 0.

Step 1 - Determine the form

Identify the form of the limit expression and what happens when \(x\) and \(y\) approach \(0\). In this case, both the numerator \(xy\) and the denominator \(\sqrt{x^2 + y^2}\) tend to \(0\) as \(x\) and \(y\) approach \(0\). The expression is indeterminate of the form \frac{0}{0}\.

Step 2 - Convert to polar coordinates

To analyze the limit more easily, convert the expression to polar coordinates where \(x = r \cos \theta\) and \(y = r \sin \theta\). This converts the original expression to \frac{r^2 \cos \theta \sin \theta}{r} = r \cos \theta \sin \theta\. Now, analyze this expression as \(r ightarrow 0\).

Step 3 - Evaluate the limit

The expression \(r \cos \theta \sin \theta\) simplifies to \(r (rac{1}{2} \sin 2\theta)\). As \(r ightarrow 0\), the entire expression approaches 0, regardless of the value of \(\theta\).

Step 4 - Conclude limit existence

Since, as \(r ightarrow 0\), the value of \(r \cos \theta \sin \theta\) approaches \(0\) for any angle \(\theta\), the limit is the same along any path toward \(0, 0\). Therefore, the limit exists and is equal to 0.

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points on a plane using a distance and an angle. Unlike Cartesian coordinates, which use \(x\) and \(y\) values, polar coordinates work with \(r\) (the radial distance from the origin) and \(\theta\) (the angle from the positive x-axis).
You can convert from Cartesian to polar coordinates using the formulas:

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

These conversions are helpful in simplifying certain calculus problems, like finding limits, where the behavior of functions is easier to understand in terms of distances and angles.
When points approach the origin, it means \(r\) is getting closer to zero. Polar coordinates often simplify expressions, especially in multivariable calculus problems.

Limit in Two Variables

Calculating limits in two variables, such as \(x\) and \(y\), involves finding how a function behaves as both variables approach a particular point. This is more complex than limits in one dimension since there are infinitely many paths to consider.
The key challenge is to ensure that the function's value approaches the same number regardless of the path taken. For example, approaching along the x-axis means \(y = 0\), while the y-axis gives \(x = 0\).
In this exercise, we check the limit by transforming to polar coordinates, considering distances and angles rather than x and y steps. The goal is for the limit to result in the same value along all paths, confirming its existence.

Indeterminate Forms

Indeterminate forms are expressions that do not initially lead to a clear limit. For example, \(\frac{0}{0}\) is a classic indeterminate form. It suggests that the function might approach any value because both the numerator and the denominator shrink to zero.
In this exercise, converting to polar coordinates transforms the fraction \(\frac{xy}{\sqrt{x^2 + y^2}}\) to \(r \cos \theta \sin \theta\). Here, examining the behavior of \(r\) as it approaches zero helps resolve the indeterminate form.
Indeterminate forms motivate alternative strategies, like changing coordinates or applying L'Hôpital's Rule, to better evaluate limits without ambiguity. Understanding these forms is critical in handling complex limit problems in calculus.

Limits in Polar Coordinates

Limits in polar coordinates provide a useful method for evaluating limits in multivariable calculus. Converting expressions using radius and angle can simplify calculations, making difficult problems more manageable.
For our given exercise, the transformation gave \(r \cos \theta \sin \theta\) from the original expression. Evaluating this limit involves understanding how each component behaves as \(r \) approaches zero.
The expression \(r(\frac{1}{2}\sin 2\theta)\) shows how \(r\) drives the entire expression toward zero since the trigonometric part \(\frac{1}{2}\sin 2\theta\) is finite. This approach works well for functions where conversion simplifies path considerations, showing the limit is consistent for all \(\theta\).
This technique exemplifies the versatility of polar coordinates in simplifying multivariable limits, emphasizing the consistency across different paths to the origin.