Problem 60 - Calculus Early Transcendentals

Question

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(0,0)} \frac{y^{2}}{x^{8}+y^{2}}$$

Step-by-Step Solution

Answer

Answer: The limit does not exist.

Step 1: Interpret the problem

We need to find the limit of the following expression as (x, y) approaches (0, 0): $$\lim _{(x, y) \rightarrow(0,0)} \frac{y^{2}}{x^{8}+y^{2}}$$

Step 2: Check in which direction to approach(0, 0)

In order to examine if the limit exists, we need to check the behavior of the function when approaching (0, 0) along different paths. Let's start by approaching along the x-axis and the y-axis: 1. Along the x-axis (y = 0): $$\lim_{x \rightarrow 0} \frac{0^{2}}{x^{8}+0^{2}} = \lim_{x \rightarrow 0} \frac{0}{x^{8}} = 0$$ 2. Along the y-axis (x = 0): $$\lim_{y \rightarrow 0} \frac{y^{2}}{0^{8}+y^{2}} = \lim_{y \rightarrow 0} \frac{y^{2}}{y^{2}} = 1$$ As we can observe, the limits are different along the x-axis and the y-axis. This means the limit does not exist.

Step 3: Conclusion

Since the function has different limits when approaching (0, 0) along different paths, we conclude that: $$\lim _{(x, y) \rightarrow(0,0)} \frac{y^{2}}{x^{8}+y^{2}}$$ does not exist.

Key Concepts

Limits

In multivariable calculus, limits help us understand the behavior of functions as the input approaches certain values or points. When dealing with functions of two variables, the concept of a limit becomes more involved. Instead of one-dimensional movement on the real line, we have two-dimensional movement on a plane. This means that instead of approaching from just one direction, you can approach the point from infinitely many paths. In the given problem, we needed to evaluate the limit of \( \frac{y^{2}}{x^{8}+y^{2}} \) as \((x, y)\) approaches \((0, 0)\). A key method to determine if such a limit exists is to check it along different paths leading to the point of interest. - If the limit remains the same regardless of the path, it exists.- If the limit differs along different paths, it does not exist. In this specific exercise, since the limits along the x-axis and y-axis are different, the overall limit does not exist.

Path Dependence

Path dependence is a crucial concept in evaluating limits for functions of several variables. It refers to the variation in limit values based on the path taken to approach a given point.In one-variable calculus, limits are typically path-independent because lines are approached from only two directions. However, in multivariable calculus, approaching a point along different curves or paths can lead to different outcomes.For the exercise, we examined the function \( \frac{y^{2}}{x^{8}+y^{2}} \) by taking different paths to approach the origin.

• Along the x-axis (where \(y = 0\)), the limit is 0.
• Along the y-axis (where \(x = 0\)), the limit is 1.

As these differ, it indicates that the limit does not exist. Understanding path dependence is fundamental to correctly assessing whether limits exist in multivariable functions.

Functions of Two Variables

Functions of two variables, written as \( f(x, y) \), depend on two independent variables, \(x\) and \(y\). These functions are typically represented in a three-dimensional space, where \(x\) and \(y\) are the plane coordinates, and \(f(x, y)\) represents height above that plane.These functions can exhibit much richer behavior than single-variable functions due to the added dimensionality.When analyzing such functions, it's important to remember:

• These functions can be complex and demonstrate unique features, such as peaks, valleys, and saddle points.
• Understanding how a function behaves near specific points, such as evaluating limits, often involves considering the various paths through this 2D plane.

In this particular exercise, the function \( \frac{y^{2}}{x^{8}+y^{2}} \) involved the variables \(x\) and \(y\). By studying this function’s behavior near the origin, one can gain insights into specific characteristics of multivariable functions.