Limits Of Functions In R^2 Approached From A Curve C Explained

In multivariable calculus, the concept of a limit is crucial for understanding continuity, differentiability, and other fundamental concepts. However, dealing with limits in higher dimensions, such as in $\mathbb{R}^2$, introduces complexities not encountered in single-variable calculus. One such complexity arises when considering the limit of a function as it is approached along a specific curve. This article delves into the formal definition of a limit of a function in $\mathbb{R}^2$ approached from a curve $C$, explores why this concept is important, and provides examples to illustrate its significance.

Formal Definition of a Limit Approached from a Curve

To formally define the limit of a function $f(x, y)$ as $(x, y)$ approaches a point $(a, b)$ along a curve $C$, we need to first parameterize the curve. Let $C$ be a curve in $\mathbb{R}^2$ parameterized by a continuous vector function $\mathbf{r}(t) = (x(t), y(t))$, where $t$ belongs to some interval $I \subseteq \mathbb{R}$. Suppose that as $t$ approaches a specific value $t_0$, the point $(x(t), y(t))$ approaches $(a, b)$, i.e.,

$$\lim_{t \to t_0} \mathbf{r}(t) = \lim_{t \to t_0} (x(t), y(t)) = (a, b).$$

Now, let $f(x, y)$ be a function defined on a domain $D \subseteq \mathbb{R}^2$, and let $L$ be a real number. We say that the limit of $f(x, y)$ as $(x, y)$ approaches $(a, b)$ along the curve $C$ is $L$, denoted by

$$\lim_{(x, y) \to (a, b) \atop (x, y) \in C} f(x, y) = L,$$

if for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |t - t_0| < \delta$, then $|f(x(t), y(t)) - L| < \epsilon$. In simpler terms, this means that as $t$ gets arbitrarily close to $t_0$, the values of $f(x(t), y(t))$ get arbitrarily close to $L$.

This definition is a generalization of the familiar $\epsilon-\delta$ definition of a limit from single-variable calculus. The key difference here is that we are restricting the approach to $(a, b)$ along a specific curve $C$. This restriction is crucial because, in multivariable calculus, the path taken to approach a point can significantly affect the limit's existence and value. The existence of a limit along a curve implies that as we trace the curve closer and closer to the point, the function values converge to a specific value.

Understanding this formal definition is crucial for grasping the nuances of multivariable limits. It highlights the importance of the path of approach, a concept not as prominent in single-variable calculus. The parameterization of the curve allows us to translate the two-dimensional limit problem into a one-dimensional limit problem, making it easier to analyze using familiar techniques.

Why is Approaching Along a Curve Important?

In single-variable calculus, when we consider the limit of a function $f(x)$ as $x$ approaches a point $a$, there are only two directions of approach: from the left and from the right. If the limits from both directions exist and are equal, then the limit exists. However, in multivariable calculus, the situation is far more complex. In $\mathbb{R}^2$, there are infinitely many paths along which we can approach a point $(a, b)$. These paths include straight lines, curves, spirals, and many other complex trajectories.

Path Dependence

The path dependence of limits in multivariable calculus is a critical concept. It means that the limit of a function $f(x, y)$ as $(x, y)$ approaches $(a, b)$ may depend on the path taken. In other words, the limit may exist and have one value along one curve, but exist and have a different value along another curve. It is also possible for the limit to exist along some curves but not along others. If the limit of a function depends on the path of approach, then the overall limit of the function at that point does not exist. This is a fundamental difference from single-variable calculus, where path dependence is not an issue.

The importance of considering limits along curves stems from the fact that the existence of a limit along a single path (e.g., a straight line) does not guarantee the existence of the limit in general. To prove that a limit exists, we need to show that the limit is the same along all possible paths of approach. Conversely, to prove that a limit does not exist, it suffices to find two paths along which the limits are different. This technique, known as the two-path test, is a powerful tool in multivariable calculus.

Continuity

The concept of approaching along a curve is also essential for understanding continuity in multivariable functions. A function $f(x, y)$ is continuous at a point $(a, b)$ if and only if the limit of $f(x, y)$ as $(x, y)$ approaches $(a, b)$ exists and is equal to $f(a, b)$. For this condition to hold, the limit must exist and be the same along every possible path of approach. If we only consider limits along specific curves, we might falsely conclude that a function is continuous when it is not.

Differentiability

Furthermore, the differentiability of multivariable functions is closely tied to the concept of limits along curves. The partial derivatives of a function, which are essential for defining differentiability, are themselves defined as limits. To ensure that a function is differentiable, these partial derivatives must exist and be continuous, which again requires considering limits along various paths. Differentiability in multiple dimensions builds upon the foundation of limits approached from different directions, emphasizing the significance of a comprehensive understanding of path dependence.

Examples Illustrating Path Dependence

To illustrate the importance of approaching along a curve, let's consider some examples of functions in $\mathbb{R}^2$ where the limit at a point depends on the path of approach.

Example 1: A Function with Different Limits Along Different Lines

Consider the function

$$f(x, y) = \frac{xy}{x^2 + y^2}$$

as $(x, y)$ approaches $(0, 0)$. If we approach $(0, 0)$ along the line $y = mx$, where $m$ is a constant, we have

$$\lim_{(x, y) \to (0, 0) \atop y = mx} f(x, y) = \lim_{x \to 0} \frac{x(mx)}{x^2 + (mx)^2} = \lim_{x \to 0} \frac{mx^2}{x^2(1 + m^2)} = \frac{m}{1 + m^2}.$$

The limit along the line $y = mx$ depends on the value of $m$. For instance, if $m = 0$ (approaching along the x-axis), the limit is 0. If $m = 1$ (approaching along the line $y = x$), the limit is $\frac{1}{2}$. Since the limit depends on the path of approach, the overall limit as $(x, y)$ approaches $(0, 0)$ does not exist. This example clearly demonstrates the path dependence of limits and underscores the need to consider multiple paths when evaluating limits in $\mathbb{R}^2$.

Example 2: A Function with a Limit Along Straight Lines But Not Along a Curve

Consider the function

$$f(x, y) = \frac{x^2y}{x^4 + y^2}$$

as $(x, y)$ approaches $(0, 0)$. Let's first approach $(0, 0)$ along any straight line $y = mx$. Then

$$\lim_{(x, y) \to (0, 0) \atop y = mx} f(x, y) = \lim_{x \to 0} \frac{x^2(mx)}{x^4 + (mx)^2} = \lim_{x \to 0} \frac{mx^3}{x^4 + m^2x^2} = \lim_{x \to 0} \frac{mx}{x^2 + m^2} = 0.$$

So, the limit along any straight line is 0. However, if we approach $(0, 0)$ along the curve $y = x^2$, we have

$$\lim_{(x, y) \to (0, 0) \atop y = x^2} f(x, y) = \lim_{x \to 0} \frac{x^2(x^2)}{x^4 + (x^2)^2} = \lim_{x \to 0} \frac{x^4}{x^4 + x^4} = \lim_{x \to 0} \frac{x^4}{2x^4} = \frac{1}{2}.$$

In this case, the limit along the curve $y = x^2$ is $\frac{1}{2}$, which is different from the limit along any straight line (which is 0). Therefore, the overall limit as $(x, y)$ approaches $(0, 0)$ does not exist. This example illustrates that the existence of limits along a family of curves (in this case, straight lines) does not guarantee the existence of the limit along all curves. This further reinforces the need for a thorough examination of path dependence.

Example 3: Polar Coordinates for Limit Evaluation

Sometimes, converting to polar coordinates can simplify the evaluation of limits in $\mathbb{R}^2$. Consider the function

$$f(x, y) = \frac{x^2 - y^2}{x^2 + y^2}$$

as $(x, y)$ approaches $(0, 0)$. Let $x = r\cos(\theta)$ and $y = r\sin(\theta)$. Then $x^2 + y^2 = r^2$, and as $(x, y) \to (0, 0)$, $r \to 0$. Thus,

$$\lim_{(x, y) \to (0, 0)} f(x, y) = \lim_{r \to 0} \frac{r^2\cos^2(\theta) - r^2\sin^2(\theta)}{r^2} = \lim_{r \to 0} (\cos^2(\theta) - \sin^2(\theta)) = \cos^2(\theta) - \sin^2(\theta) = \cos(2\theta).$$

The limit depends on the angle $\theta$, which represents the direction of approach. This again demonstrates that the limit depends on the path and, therefore, the overall limit does not exist. Polar coordinates provide a systematic way to analyze limits along different radial paths, making it a powerful tool for identifying path dependence.

Techniques for Determining Limits

Several techniques can be used to determine whether a limit exists and, if so, to find its value. These include:

1. Direct Substitution: If the function is continuous at the point, direct substitution may work.
2. Two-Path Test: Find two different paths along which the limits are different to show that the limit does not exist.
3. Polar Coordinates: Convert to polar coordinates to simplify the expression and analyze the limit as $r \to 0$.
4. Squeeze Theorem: If $g(x, y) \leq f(x, y) \leq h(x, y)$ and $\lim_{(x, y) \to (a, b)} g(x, y) = \lim_{(x, y) \to (a, b)} h(x, y) = L$, then $\lim_{(x, y) \to (a, b)} f(x, y) = L$.
5. Epsilon-Delta Definition: Use the formal definition to prove the limit exists and find its value.

The choice of technique depends on the specific function and the point at which the limit is being evaluated. The two-path test is particularly useful for showing that a limit does not exist, while the epsilon-delta definition is essential for rigorous proofs of limit existence.

Conclusion

In summary, understanding the limit of a function in $\mathbb{R}^2$ as it is approached along a curve is crucial in multivariable calculus. The formal definition highlights the importance of the path of approach, a concept that distinguishes multivariable limits from single-variable limits. The path dependence of limits means that the existence of a limit along one curve does not guarantee its existence along another, or in general. Examples such as $f(x, y) = \frac{xy}{x^2 + y^2}$ and $f(x, y) = \frac{x^2y}{x^4 + y^2}$ clearly illustrate this phenomenon. By understanding these concepts and utilizing techniques like the two-path test and conversion to polar coordinates, one can effectively analyze and determine limits of multivariable functions.

Ultimately, the concept of limits approached from curves forms the foundation for understanding continuity and differentiability in higher dimensions. A strong grasp of this topic is essential for anyone delving into the intricacies of multivariable calculus and its applications in various fields of science and engineering. Mastering these concepts not only enhances mathematical proficiency but also provides a deeper appreciation for the complexities and nuances of functions in multiple dimensions.