Problem 69 - 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. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}}{x^{2}+y^{2}}$$

Step-by-Step Solution

Answer

Answer: No, the limit does not exist as the expression depends on the angle θ, not only on r, and varies for different paths towards (0, 0).

Step 1: Convert to polar coordinates

To convert the given function to polar coordinates, replace \(x\) with \(r\cos{\theta}\) and \(y\) with \(r\sin{\theta}\): $$\frac{x^{2}}{x^{2}+y^{2}} = \frac{(r\cos{\theta})^2}{(r\cos{\theta})^2+(r\sin{\theta})^2}$$

Step 2: Simplify the expression

Now, simplify the expression by factoring out \(r^2\) from the denominator: $$\frac{(r\cos{\theta})^2}{(r\cos{\theta})^2+(r\sin{\theta})^2} = \frac{(r\cos{\theta})^2}{r^2(\cos^2{\theta}+\sin^2{\theta})}$$ We know that \(\cos^2{\theta}+\sin^2{\theta} = 1\), so we can further simplify the expression: $$\frac{(r\cos{\theta})^2}{r^2} = \frac{(r\cos{\theta})^2}{r^2} \cdot \frac{1}{\cos^2{\theta}+\sin^2{\theta}} = \frac{(r\cos{\theta})^2}{r^2} \cdot \frac{1}{1} = \frac{(r\cos{\theta})^2}{r^2}$$

Step 3: Evaluate the limit as \(r \to 0\)

Now that we have the expression simplified, we want to find the limit as \(r \to 0\): $$\lim_{r\to 0} \frac{(r\cos{\theta})^2}{r^2}$$ Dividing the numerator and denominator by \(r^2\), we get: $$\lim_{r\to 0} \frac{\cos^2{\theta}}{1}$$ Since the limit does not involve \(r\) anymore, the limit is equal to \(\cos^2{\theta}\). As we can see, the limit depends on the angle \(\theta\) and not only on \(r\).

Step 4: Determine whether the limit exists or not

As mentioned earlier, the limit must be the same along all paths to \((0,0)\) (i.e., for all values of \(\theta\)), but the obtained expression depends on \(\theta\). Therefore, the limit does not exist.

Key Concepts

Calculus

Calculus is a branch of mathematics that studies how things change. It's often about understanding how functions behave and their properties. In this particular problem, we are dealing with the limit of a function of two variables. The expression \[\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}}{x^{2}+y^{2}}\]is a function that describes changes in two variables, \(x\) and \(y\), as they move towards zero. Calculus provides us with tools like derivatives and integrals, but here we focus on limits. Limits help us understand the behavior of a function as the input approaches a certain point. The concept of limits let's us predict what the value of a function will be as its variables get exceedingly close to a particular point. In this exercise, we use limits to explore how the function behaves as both \(x\) and \(y\) approach zero.

Limits

Limits describe what happens to a function as the input gets close to a point, but does not actually reach it. They're crucial for determining continuity and behavior of functions.When evaluating \[\lim_{(x, y) \rightarrow(0,0)} \frac{x^{2}}{x^{2}+y^{2}}\]we want the same limit to be obtained regardless of the path taken to \(0,0\). If this is not true, the limit does not exist. In our solution, we transformed the Cartesian coordinates \(x, y\) into polar coordinates \(r, \theta\) because it's often easier to evaluate limits as \(r \rightarrow 0\) than directly approaching \(0,0\) in the original coordinates. The polar transformation simplifies handling the limit because it puts all paths to zero into a single parameter, \(r\), which makes checking different paths straightforward. If \(r\) can approach zero through all such paths and result in the same limit, then that limit exists. However, as seen in this exercise, the determined limit depends on \(\theta\), the angle, leading us to conclude that the limit does not exist uniformly.

Coordinate Transformation

Coordinate transformation is the process of changing a given set of coordinates to another one to solve a problem more easily. Different coordinate systems offer advantages in analyzing the same problem in diverse contexts. In this problem, Cartesian coordinates, which describe a point in\((x,y)\) can be transformed into polar coordinates \((r, \theta)\). This transformation is done through the relationships:

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

By changing to polar coordinates, the problem of finding a limit becomes simpler because we get to focus on the parameter \(r\) alone as it approaches zero, essentially reducing the complexity. This allows the function to be expressed as \(\frac{(r\cos{\theta})^2}{r^2}\), simplifying the evaluation of how the function behaves as \(r\) becomes nearly zero. However, as we evaluated, the presence of \(\theta\) in the expression led to a dependency on the angle path, showing us that coordinate transformation can also help identify such path-dependent behavior in limits.