Problem 71 - Calculus Early Transcendentals

Question

Limits at (0,0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as \(r \rightarrow 0\) along all paths to (0,0) Evaluate the following limits or state that they do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{(x-y)^{2}}{\left(x^{2}+y^{2}\right)^{3 / 2}}$$

Step-by-Step Solution

Answer

Answer: No, the limit does not exist.

Step 1: Convert to polar coordinates

In polar coordinates, we have \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). We will substitute these expressions for \(x\) and \(y\) in the given limit. So, $$ \lim _{(x, y) \rightarrow(0,0)} \frac{(x-y)^{2}}{\left(x^{2}+y^{2}\right)^{3 / 2}} = \lim _{(r,\theta) \rightarrow(0,\theta)} \frac{(r\cos(\theta)-r\sin(\theta))^2}{(r^2\cos^2(\theta)+r^2\sin^2(\theta))^{\frac{3}{2}}} $$

Step 2: Simplify the expression

The expression can be simplified as follows: $$ \lim _{(r,\theta) \rightarrow(0,\theta)} \frac{(r\cos(\theta)-r\sin(\theta))^2}{(r^2\cos^2(\theta)+r^2\sin^2(\theta))^{\frac{3}{2}}} = \lim _{(r,\theta) \rightarrow(0,\theta)} \frac{r^2(\cos^2(\theta)-2\cos(\theta)\sin(\theta)+\sin^2(\theta))}{r^3(\cos^2(\theta)+\sin^2(\theta))^{\frac{3}{2}}} $$ Now, we can cancel one \(r\) from the numerator and the denominator, and also we can use the trigonometric identity \(\cos^{2}(\theta)+\sin^{2}(\theta) = 1\) to further simplify: $$ \lim_{(r,\theta) \rightarrow(0,\theta)} \frac{(\cos^2(\theta)-2\cos(\theta)\sin(\theta)+\sin^2(\theta))}{r^2} $$

Step 3: Evaluate the limit

Now, we will evaluate the limit as \(r \rightarrow 0\): $$ \lim_{(r,\theta) \rightarrow(0,\theta)} \frac{(\cos^2(\theta)-2\cos(\theta)\sin(\theta)+\sin^2(\theta))}{r^2} = \frac{\cos^2(\theta)-2\cos(\theta)\sin(\theta)+\sin^2(\theta)}{0} $$ As we can see, the expression is indeterminate in this form. Let's try to take the limit as \(r \rightarrow 0^+\): $$ \lim_{(r,\theta) \rightarrow(0^+,\theta)} \frac{(\cos^2(\theta)-2\cos(\theta)\sin(\theta)+\sin^2(\theta))}{r^2} = \frac{\cos^2(\theta)-2\cos(\theta)\sin(\theta)+\sin^2(\theta)}{0^+} $$ This expression is also indeterminate. Since the limit is indeterminate along different paths to (0,0), we conclude that the limit does not exist.

Key Concepts

Polar Coordinates

In the realm of mathematics, polar coordinates present an alternative to the standard Cartesian coordinates for describing positions in a plane. Instead of using x and y to specify a point, polar coordinates utilize a distance from the origin, denoted as \( r \), and an angle, \( \theta \), measured from the positive x-axis.

To convert Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\), we use the transformations:

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

By expressing the limit in the problem using polar coordinates, we're able to reframe the computation in terms that often simplify the problem. In polar coordinates, the distance between points is neatly represented by \( r \), and the direction is handled by \( \theta \), making them ideal for resolving limits at the origin (0,0).

Indeterminate Forms

In calculus, an indeterminate form arises when substituting a value into a function results in an undefined expression. The most common indeterminate forms include \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( \infty - \infty \), and others. These forms signal the need for further analysis rather than yielding a straightforward numerical value.

In the given problem, the application of polar coordinates simplifies the expression to a form that, when \( r \) approaches zero, remains indeterminate (since it leads to division by zero). Indeterminate forms often require techniques such as L'Hôpital's Rule, algebraic simplification, or substitution to resolve. In this particular exercise, despite multiple path evaluations and simplifications, the limit remains undefined, suggesting the limit does not exist.

Multivariable Calculus

Multivariable calculus extends single-variable calculus concepts to higher dimensions. It involves functions with more than one variable, such as \( f(x, y) \), and often explores properties and computations related to limits, derivatives, and integrals in these expanded settings.

The problem at hand is a classic multivariable calculus exercise, which involves finding the limit of a function as both \( x \) and \( y \) simultaneously approach a particular point, in this case, (0,0). Problems like this require techniques that are unique to higher-dimensional spaces, such as considering the limit along multiple paths and sometimes converting to polar coordinates for simplification. As seen here, depending on the approach and the function’s behavior along these paths, the limit might not exist if it doesn't resolve consistently.

Trigonometric Identities

Trigonometric identities are critical tools in both calculus and algebra for simplifying and resolving complex expressions. The Pythagorean identity, \( \cos^2(\theta) + \sin^2(\theta) = 1 \), is one of the most fundamental and is prominently used in the simplification of polar-coordinate expressions.

In the solution provided, we leverage this identity to simplify the denominator of the limit expression. Simplifying using identities allows us to reduce expressions to forms that are easier to work with or evaluate. Correctly applying these identities not only streamlines computations but can also reveal whether further transformations are necessary for understanding the behavior of functions as variables approach certain values.