Problem 22 - Calculus with Concepts in Calculus

Question

Use the definition of \(\lim _{(x, y)_{\vec{R}}\left(x_{0}, y_{0}\right)} f(x, y)\) to determine whether the given limit exists for the given region \(R\). If the limit exists, find it. \(\lim _{(x, y) \vec{R}(0,0)} x e^{-1 / y \mid} ; R\) consists of all \((x, y)\) such that \(y \neq 0\)

Step-by-Step Solution

Answer

The limit exists and is 0.

Step 1: Understand the problem context

We need to determine whether the given limit \( \lim_{(x, y) \to (0,0)} x e^{-1/|y|} \) exists over a specified region \( R \), which includes all points \((x, y)\) where \( y eq 0 \).

Step 2: Evaluate directional limits

In a two-variable limit problem, we need to check the limit from different paths towards the point \((0,0)\). Common paths include approaching along the x-axis \((0,y)\) and y-axis \((x,0)\), as well as other paths such as \( y = mx \).

Step 3: Limit along x-axis \((y=constant)\)

First consider the path \( y = c \), where \(c\) is any non-zero constant. The expression becomes \( \lim_{x \to 0} x e^{-1/|c|} \). This simplifies to \( 0 \times e^{-1/|c|} = 0 \).

Step 4: Limit along y-axis \((x=constant)\)

Now consider the path \( x = 0 \) and let \( y \to 0 \). The limit becomes \( \lim_{y \to 0} 0 \times e^{-1/|y|} = 0 \).

Step 5: Limit along arbitrary paths \((y=mx)\)

Consider the path \( y = mx \), where \( m eq 0 \). The expression is \( \lim_{x \to 0} x e^{-1/|mx|} \). As \( x \to 0 \), the exponential term grows very large, but since it's multiplied by \( x \), the result is \( 0 \).

Step 6: Conclude existence of the limit

Since the limit approaches 0 along the common paths we tested and \( y eq 0 \) in region \( R \), the limit \( \lim_{(x, y) \to (0,0)} x e^{-1/|y|} \) exists and equals 0.

Key Concepts

Directional Limits

In the realm of multivariable calculus, determining the limit of a function as it approaches a point often involves exploring different paths toward that point. These paths are referred to as directional limits. When dealing with two-variable functions, it's crucial to evaluate the limit from various directions to ensure consistency.
For the problem at hand, we examine the directional limits of several paths to determine if the function converges to the same value when approaching the origin, \((0,0)\).
Some common paths include:

• The x-axis, where we hold the y-coordinate constant as y = 0 and allow x to approach zero.
• The y-axis, where we hold the x-coordinate constant as x = 0, letting y approach zero.
• Diagonal and arbitrary paths, such as lines where y = mx for some non-zero slope \( m \).

Evaluating along these paths, we consistently found that the limit of the function as it approaches \((0,0)\) is \( 0 \). Notably, unless all directional limits coincide to a single value, the overall limit may not exist.

Existence of Limits

The existence of a limit—especially for functions of two variables—requires that the function exhibits a consistent behavior from every conceivable direction as it approaches a given point. For a limit to exist at a point \((x_0, y_0)\), the function should provide the same outcome from every path leading to that point.
In simpler terms, despite taking any path towards our target \((x_0, y_0)\), the result should remain the same. If any one path gives a different result, the overall limit does not exist.
In the exercise above, our exploration of various paths to the origin, such as the x-axis, y-axis, and a general y = mx path, confirmed the limit as \(0\) consistently. Because the function behaved uniformly across multiple directional approaches, we can confidently assert that the limit exists.
Uniformity of results reinforces the conclusion that the directional limit represents the true limit, confirming the function's consistency and predictability at the point in question.

Two-variable Functions

A two-variable function, such as the one given in our exercise, involves inputs from two dimensions. Each variable, \( x \) and \( y \), influences the function outcome. This dual-dependence forms surfaces in three-dimensional space, complicating our quest to deduce how the function behaves near specific points.
When focus shifts to the concept of limits for these functions, the multivariable aspect necessitates a deliberation of approaches along various axes and lines in the two-variable plane. Unlike single-variable calculus—where approaching from the left or right suffices—two-variable calculus demands more thorough scrutiny.
The challenge often lies in analyzing each path to ensure the function behaves consistently. Our task, then, involves solving \( \lim_{(x, y) \to (0,0)} x e^{-1/|y|} \) by extensively testing and deducing from these different directions. By using paths like y-constant (x-axis), x-constant (y-axis), and arbitrary linear paths (y=mx), we've determined the function's behavior: consistently yielding the same limit value.
The exercise teaches that while two-variable functions may initially seem daunting, involving additional dimensions only enriches our analysis, providing more detail and precision in understanding the nature of such complex systems.