Problem 70 - 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. $$\lim _{(x, y) \rightarrow(0,0)} \frac{(x-y)^{2}}{x^{2}+x y+y^{2}}$$

Step-by-Step Solution

Answer

Question: Evaluate the limit as (x, y) approaches (0, 0) of the expression: $$\frac{(x-y)^{2}}{x^{2}+x y+y^{2}}$$ Answer: The limit as (x, y) approaches (0, 0) of the given expression is: $$\frac{(\cos^2\theta - 2\cos\theta\sin\theta + \sin^2\theta)}{(\cos^{2}\theta + \cos\theta \sin\theta + \sin^{2}\theta)}$$

Step 1: Convert to polar coordinates

First, let us substitute \(x = r \cos\theta\) and \(y = r \sin\theta\) in the given expression: $$\frac{(x-y)^{2}}{x^{2}+x y+y^{2}} = \frac{(r \cos\theta - r \sin\theta)^{2}}{r^{2} \cos^{2}\theta + r^{2} \cos\theta \sin\theta + r^{2} \sin^{2}\theta}$$

Step 2: Simplify the expression

Now, let's simplify the expression: $$\frac{(r (\cos\theta - \sin\theta))^{2}}{r^{2} (\cos^{2}\theta + \cos\theta \sin\theta + \sin^{2}\theta)}$$

Step 3: Factor out common terms

We can factor out \(r^2\) from both the numerator and the denominator: $$\frac{r^2 (\cos^2\theta - 2\cos\theta\sin\theta + \sin^2\theta)}{r^2 (\cos^{2}\theta + \cos\theta \sin\theta + \sin^{2}\theta)}$$ $$\frac{(\cos^2\theta - 2\cos\theta\sin\theta + \sin^2\theta)}{(\cos^{2}\theta + \cos\theta \sin\theta + \sin^{2}\theta)}$$

Step 4: Evaluate the limit

We need to evaluate the limit as \(r \rightarrow 0\). However, as r does not appear in the expression, there is no need to take the limit with respect to r. Instead, we need to check if the same value is obtained for all the possible paths to \((0,0)\). This means that we need to check if the expression depends on \(\theta\). Since the expression only contains \(\cos\theta\) and \(\sin\theta\), and not independent \(\theta\)s, we can say that the limit exists and it's the same as the expression itself: $$\lim _{(x, y) \rightarrow(0,0)} \frac{(x-y)^{2}}{x^{2}+x y+y^{2}} = \frac{(\cos^2\theta - 2\cos\theta\sin\theta + \sin^2\theta)}{(\cos^{2}\theta + \cos\theta \sin\theta + \sin^{2}\theta)}$$

Key Concepts

Limits

In calculus, a limit is a value that a function approaches as the input approaches some value. When dealing with functions of two variables, such as in the exercise, the concept of limits requires us to find the value a function gets closer to as both variables approach a particular point. In this case, the point is \((0, 0)\). For multivariable functions, confirming the existence of a limit requires checking whether the limit is consistent across all possible paths to that point.
Here, converting to polar coordinates and simplifying the expression lets us evaluate the limit by checking its dependence on \(\theta\) instead of directly focusing on \(x\) and \(y\). By doing this, we ensure consistency across different paths to \(0, 0\).
Since the final simplified expression doesn't depend on \((r, \theta)\) except for \(\cos\) and \(\sin\), showing that the value of the limit is independent of the path and therefore exists.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions of several variables. This involves handling limits, derivatives, and integrals in higher dimensions. In single-variable calculus, you deal with paths on a line, but with multivariable functions, you operate in a plane or space, requiring more advanced techniques to evaluate limits or other properties.
When calculating a limit in multivariable calculus, ensure that the result is consistent no matter from which direction you approach the target point. For instance, moving along various paths like straight lines or parabolic curves should all yield the same outcome if the limit truly exists. This increased complexity necessitates new approaches, such as using polar coordinates, which simplify determining the limits of multivariable functions.
With polar coordinates, conversion reduces the two-variable problem to one involving parameters \(r\) and \(\theta\), making identification of the limit more straightforward and highlighting the importance of path-independence in verifying multivariable limits.

Coordinate Transformation

Coordinate transformation is a method used to simplify complex problems by changing the coordinate system under consideration. In the exercise, transforming Cartesian coordinates (\(x, y\)) into polar coordinates (\(r, \theta\)) makes the limit problem more manageable.

• The Cartesian system uses right-angled coordinates, which are intuitive and straightforward for problems aligned with these axes.
• Polar coordinates, represented by \(r\) for distance from the origin and \(\theta\) for the angle, are beneficial for problems involving symmetrical properties or rotational movements.

By switching to polar coordinates, we focus on the path to the origin through the radial parameter \(r\) and angular parameter \(\theta\), simplifying limit determination. This transformation demonstrates the flexibility and utility of using alternative coordinate systems to tackle complex multidimensional calculus problems effectively.