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 pains in the domain to \((0, 0).\) Evaluate the following limits or state that they do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}+x^{2} y^{2}}{x^{2}+y^{2}}$$

Short answer

Based on the step by step solution, the limit of the given function as it approaches (0,0) exists and is equal to 1.

Step-by-step solution

Convert the function to polar coordinates

Recall the relationships between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): $$x = r\cos{\theta}, \,\, y = r\sin{\theta}$$ We'll convert the given function using these relationships: $$f(x, y) = \frac{x^{2}+y^{2}+x^{2} y^{2}}{x^{2}+y^{2}} = \frac{r^{2}\cos^{2}{\theta} + r^{2}\sin^{2}{\theta} + r^{4}\cos^{2}{\theta}\sin^{2}{\theta}}{r^{2}\cos^{2}{\theta} + r^{2}\sin^{2}{\theta}}$$

Simplify the function

Now let's simplify the expression: $$\frac{r^{2}\cos^{2}{\theta} + r^{2}\sin^{2}{\theta} + r^{4}\cos^{2}{\theta}\sin^{2}{\theta}}{r^{2}\cos^{2}{\theta} + r^{2}\sin^{2}{\theta}} = \frac{r^{2}(\cos^{2}{\theta}+\sin^{2}{\theta}) + r^{4}\cos^{2}{\theta}\sin^{2}{\theta}}{r^{2}(\cos^{2}{\theta}+\sin^{2}{\theta})}$$ Since \(\cos^{2}{\theta}+\sin^{2}{\theta}=1\), the expression simplifies to: $$\frac{r^{2} + r^{4}\cos^{2}{\theta}\sin^{2}{\theta}}{r^{2}}$$

Evaluate the limit

Now let's try to evaluate the limit as \((r,\theta) \rightarrow (0,0)\): $$\lim _{(r, \theta) \rightarrow(0,0)} \frac{r^{2} + r^{4}\cos^{2}{\theta}\sin^{2}{\theta}}{r^{2}}$$ We can cancel out \(r^2\) in the numerator and denominator: $$\lim _{(r, \theta) \rightarrow(0,0)} (1 + r^{2}\cos^{2}{\theta}\sin^{2}{\theta})$$ Since \(r^2\cos^{2}{\theta}\sin^{2}{\theta} \ge 0\), we can say that: $$1 \le 1 + r^2\cos^2{\theta}\sin^2{\theta} \le 1+r^2$$ Now as \(r \rightarrow 0\), we have: $$1 \le 1 + r^2\cos^2{\theta}\sin^2{\theta} \le 1$$ By the squeeze theorem, we can conclude that: $$\lim _{(r, \theta) \rightarrow(0,0)} (1 + r^{2}\cos^{2}{\theta}\sin^{2}{\theta}) = 1$$ So the limit exists and is equal to \(1\).

Key concepts

Polar Coordinates

Understanding polar coordinates is fundamental in various branches of mathematics, especially when dealing with problems involving curves and regions in the plane. Polar coordinates provide an alternative to the traditional Cartesian (x, y) coordinate system. Instead of using horizontal and vertical distances, polar coordinates measure the distance from a central point, known as the origin, and the angle from a reference direction, typically the positive x-axis.

Formally, a point in polar coordinates is represented as \(r, \theta\), where \(r\) is the radius or the distance to the point from the origin, and \(\theta\) is the angle measured in radians from the reference direction (counter-clockwise is positive). This system is particularly useful when dealing with circular and spiral patterns, among other things.

In the context of limits and calculus, polar coordinates often simplify the process of finding limits of functions that are not easily handled in Cartesian coordinates.

Squeeze Theorem

The Squeeze Theorem, also known as the Sandwich Theorem, is a crucial concept when discussing limits in calculus. It provides a way to find the limits of functions that might not be easily computed directly.

The core idea is that if a function \(f(x)\) is \(\text{always}\) bounded between two other functions \(g(x)\) and \(h(x)\), such that \(g(x) \le f(x) \le h(x)\), and if the limits of \(g(x)\) and \(h(x)\) as \(x\) approaches a certain value are the same, then the limit of \(f(x)\) must also approach that same value.

In other words, \(f(x)\) is 'squeezed' into a specific value at a point. This can make determining the limit of complex functions possible, even in scenarios where the function itself seems to have indeterminate behavior at a specific point.

Limits of Functions

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as its argument approaches a particular point. It doesn't necessarily have to equal the function's value at that point, but instead what value it \(\text{seems}\) to approach as the input gets arbitrarily close to that point.

Limits can be used to describe the continuity of a function, its derivative, and the behavior of functions near singularities or infinity. The concept of limits can often seem abstract, but it allows mathematicians and scientists to make precise the idea of very gradual change, and to talk about things like 'instantaneous' rates of change in the context of derivatives.

When limits fail to exist, it can indicate an intrinsic 'jump' or discontinuity in the function or an oscillation that does not resolve to a single value as one approaches the limit point.

Cartesian to Polar Conversion

Converting between Cartesian coordinates and polar coordinates is often necessary in mathematics, particularly in fields like physics, engineering, and computer graphics, where various coordinate systems are used depending on the problem at hand.

The conversion is relatively straightforward and relies on the relationship between angles and distances: the x-coordinate can be found by multiplying the radius \(r\) by the cosine of the angle \(\theta\), and the y-coordinate by multiplying the radius by the sine of the angle: \[ x = r \cos(\theta), \quad y = r \sin(\theta) \.\] Conversely, the radius can be found as the square root of the sum of the squares of the x and y coordinates, and the angle can be determined by taking the arctangent of the ratio of y to x.

This type of conversion is particularly useful when working on problems that are inherently circular or radial, as it can significantly simplify the computations and visual understanding of the problem.