Problem 19 - APEX Calculus

Question

Evaluate the limit along the paths given, then state why these results show the given limit does not exist. \(\lim _{(x, y) \rightarrow(1,2)} \frac{x+y-3}{x^{2}-1}\) (a) Along the path \(y=2\). (b) Along the path \(y=x+1\).

Step-by-Step Solution

Answer

The limit does not exist since along different paths the limit values differ: \( \frac{1}{2} \) for \( y=2 \) and 1 for \( y=x+1 \).

Step 1: Evaluate the limit along the path y=2

Along the path \( y = 2 \), substitute \( y = 2 \) into the function. The function becomes \( \frac{x + 2 - 3}{x^2 - 1} = \frac{x - 1}{x^2 - 1} \). Notice that this is the same as \( \frac{x - 1}{(x-1)(x+1)} \) provided \( x eq 1 \). Simplifying, we get \( \frac{1}{x + 1} \). As \( x \rightarrow 1 \), \( \frac{1}{x+1} \rightarrow \frac{1}{1+1} = \frac{1}{2} \). Thus, the limit along this path is \( \frac{1}{2} \).

Step 2: Evaluate the limit along the path y=x+1

Along the path \( y = x + 1 \), substitute \( y = x + 1 \) into the function. The function becomes \( \frac{x + (x + 1) - 3}{x^2 - 1} = \frac{2x - 2}{x^2 - 1} \). Notice that this is the same as \( \frac{2(x-1)}{(x-1)(x+1)} \) provided \( x eq 1 \). Simplifying, we get \( \frac{2}{x + 1} \). As \( x \rightarrow 1 \), \( \frac{2}{x+1} \rightarrow \frac{2}{1+1} = 1 \). Thus, the limit along this path is \( 1 \).

Step 3: Conclude non-existence of the limit

Since the limit along the path \( y = 2 \) is \( \frac{1}{2} \) and along the path \( y = x + 1 \) it is \( 1 \), these two different limits demonstrate that the overall limit \( \lim _{(x, y) \rightarrow(1,2)} \frac{x+y-3}{x^{2}-1} \) does not exist. In multivariable calculus, for a limit to exist at a point, it must have the same value regardless of the path taken to reach that point.

Key Concepts

Path-Dependent Limits

When analyzing multivariable limits, it's crucial to understand path-dependent limits. Unlike functions of a single variable, where approaching from either direction on the number line yields the same result, multivariable functions offer multiple ways to approach a point. This characteristic makes it essential to investigate various paths.
In our exercise, evaluating the limit \(\lim _{(x, y) \rightarrow(1,2)} \frac{x+y-3}{x^{2}-1}\) means considering different paths that lead to the point \((1, 2)\).
We explored two paths:

• The path \(y = 2\) yielded \(\frac{1}{2}\).
• The path \(y = x + 1\) gave us \(1\).

These differing results underscore that the function behaves differently depending on how it is approached, a key indication of path-dependency. Path-dependent behavior can be challenging but is a central aspect of understanding multivariable limits.

Limit Non-Existence

A limit does not exist in the multivariable context if approaching the point through different paths results in varying values. This exercise illustrates this concept perfectly. Two distinct results from our paths — \(\frac{1}{2}\) along \(y = 2\) and \(1\) along \(y = x + 1\) — indicate that the limit cannot settle on a single value.
In simpler terms, for a multivariable limit to exist at a point, all paths leading to that point must converge to the same limit value. If even one path yields a different result, the overall limit is deemed undefined, or non-existent. This disparity in path results led to the conclusion that \(\lim _{(x, y) \rightarrow(1,2)} \frac{x+y-3}{x^{2}-1}\)does not exist.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus beyond functions of a single variable to functions depending on several variables. This branch addresses the challenge of dealing with seemingly limitless pathways in real coordinate planes and spaces.
In this context, limits are more complex. Functions are no longer constrained to the real number line but are expanded to planes, surfaces, and spaces. Evaluating these multivariable limits requires understanding paths, directions, and behaviors of functions. Often, confirming path-independence is a crucial test to affirm the existence of such a limit.

• Multiple paths: Approach points via various routes.
• Path independence: A single limit must hold true across different paths.
• Conceptual understanding: Limits, continuity, and differentiability extend to higher dimensions.

This exercise is a practical application of multivariable calculus, sharpening skills in determining when multivariable limits do or do not exist.