Evaluating The Limit Of (x+y-4)/(√(x+y)-2) As (x, Y) Approaches (2, 2) With X+y=4

In the realm of multivariable calculus, evaluating limits of functions with multiple variables often presents a unique set of challenges. Unlike single-variable calculus, where we approach a point along a single axis, multivariable limits require us to consider paths approaching the point from various directions. This complexity increases significantly when a constraint is imposed on the variables, restricting the possible paths of approach. This article delves into the intricacies of evaluating such limits, focusing on a specific example that highlights key techniques and concepts.

Understanding the Limit Definition in Multivariable Calculus

Before we tackle the specific problem, it's crucial to understand the fundamental definition of a limit in the context of multivariable calculus. The limit of a function f(x, y) as (x, y) approaches a point (a, b) exists and equals L if, for every ε > 0, there exists a δ > 0 such that |f(x, y) - L| < ε whenever 0 < √(x - a)² + (y - b)² < δ. In simpler terms, this means that we can make the function's value arbitrarily close to L by making the point (x, y) sufficiently close to (a, b), but not equal to (a, b). The challenge arises when a constraint is placed on the variables, such as x + y = 4 in our example. This constraint limits the possible paths of approach, requiring a different strategy for evaluation.

When dealing with constrained limits, the key is to incorporate the constraint into the function, effectively reducing the problem to a single-variable limit or a simpler multivariable limit. This can be achieved through substitution or parameterization, allowing us to analyze the function's behavior along the specified constraint. In our case, the constraint x + y = 4 allows us to express one variable in terms of the other, simplifying the expression and making the limit evaluation more manageable.

The Importance of Path Independence

A critical concept in multivariable limits is path independence. For a limit to exist, the function must approach the same value L regardless of the path taken to approach the point (a, b). If different paths lead to different limit values, the limit does not exist. This is a major distinction from single-variable calculus, where there are only two paths of approach (from the left and from the right). In the context of constraints, path independence means that the limit must be the same along all paths that satisfy the constraint. If we find a single path along the constraint that yields a different limit, we can conclude that the overall limit does not exist.

Problem Statement and Initial Analysis

Let's consider the limit:

$$\lim _{\substack{x, y \rightarrow(2,2) \ x+y=4}} \frac{x+y-4}{\sqrt{x+y}-2}$$

This limit asks us to find the value that the function f(x, y) = (x + y - 4) / (√(x + y) - 2) approaches as (x, y) approaches the point (2, 2), subject to the constraint x + y = 4. A direct substitution of x = 2 and y = 2 into the function results in the indeterminate form 0/0, indicating that further analysis is required. This indeterminate form is a common signal in limit problems, suggesting the need for algebraic manipulation or a different approach to evaluate the limit.

The constraint x + y = 4 plays a crucial role in this problem. It restricts the possible values of x and y, forcing them to lie on a line in the xy-plane. This means that we are not approaching (2, 2) from all directions, but only along the line defined by x + y = 4. This constraint simplifies the problem, as we can use it to eliminate one of the variables and reduce the limit to a single-variable problem.

Applying the Constraint and Simplifying the Expression

The most straightforward approach to handle the constraint x + y = 4 is to substitute it directly into the function. Replacing x + y with 4 in the numerator, we get x + y - 4 = 4 - 4 = 0. Similarly, in the denominator, we have √(x + y) - 2 = √4 - 2 = 2 - 2 = 0. This confirms that we indeed have the indeterminate form 0/0. However, the substitution has also simplified the numerator to 0, which is a key observation.

With the constraint applied, the function becomes:

f(x, y) = 0 / (√(x + y) - 2)

Since the numerator is 0 and the denominator is not identically zero in the neighborhood of (2, 2) along the constraint x + y = 4, the function simplifies to 0. This means that the limit is likely to be 0, but we need to rigorously verify this.

To further solidify our understanding, we can perform algebraic manipulation to eliminate the indeterminate form. One common technique for dealing with square roots in the denominator is to multiply both the numerator and the denominator by the conjugate of the denominator. In this case, the conjugate of √(x + y) - 2 is √(x + y) + 2. Multiplying both the numerator and the denominator by this conjugate, we get:

f(x, y) = [(x + y - 4) * (√(x + y) + 2)] / [(√(x + y) - 2) * (√(x + y) + 2)]

Simplifying the denominator using the difference of squares formula, we have:

f(x, y) = [(x + y - 4) * (√(x + y) + 2)] / [(x + y) - 4]

Now, we can cancel the (x + y - 4) terms in the numerator and denominator, provided that x + y ≠ 4. However, since we are considering the limit as (x, y) approaches (2, 2) along the constraint x + y = 4, this condition is not strictly met. Nevertheless, we can consider values of (x, y) that are arbitrarily close to (2, 2) but not exactly equal, allowing us to perform the cancellation. This subtle point is crucial in understanding the rigor of limit evaluation.

After cancellation, we are left with:

f(x, y) = √(x + y) + 2

Evaluating the Simplified Limit

Now that we have simplified the function, we can evaluate the limit by substituting the constraint x + y = 4 into the simplified expression:

lim (x, y)→(2, 2), x+y=4 [√(x + y) + 2] = √4 + 2 = 2 + 2 = 4

This result seems contradictory to our earlier intuition that the limit should be 0. This discrepancy highlights the importance of careful algebraic manipulation and the potential pitfalls of premature conclusions. The error lies in our assumption that the function simplifies to 0 after the initial substitution. While the numerator becomes 0, the denominator also becomes 0, leading to the indeterminate form. The subsequent algebraic manipulation was necessary to correctly evaluate the limit.

Alternative Approach: Single Variable Limit

Another way to approach this problem is to use the constraint x + y = 4 to express one variable in terms of the other. For example, we can write y = 4 - x. Substituting this into the function, we obtain a single-variable limit:

$$\lim _{x \rightarrow 2} \frac{x+(4-x)-4}{\sqrt{x+(4-x)}-2} = \lim _{x \rightarrow 2} \frac{0}{\sqrt{4}-2} = \lim _{x \rightarrow 2} \frac{0}{0}$$

This again results in the indeterminate form 0/0, but it is now a single-variable limit, which we can handle using techniques from single-variable calculus. However, notice that the expression simplifies immediately to 0 in the numerator. This indicates a potential issue with direct substitution and calls for further analysis.

Instead, let’s substitute y = 4 - x into the simplified form of the function we obtained earlier:

f(x) = √(x + (4 - x)) + 2 = √4 + 2 = 4

Now, the limit is simply:

lim x→2 [4] = 4

This confirms our earlier result obtained through algebraic manipulation and reinforces the importance of simplifying the expression before evaluating the limit.

Conclusion: Mastering Constrained Limits

In conclusion, evaluating limits of multivariable functions under constraints requires a careful blend of algebraic manipulation, substitution, and a deep understanding of the limit definition. The presence of a constraint restricts the paths of approach, often simplifying the problem but also introducing potential pitfalls if not handled correctly. In the specific example we explored, the limit of the function f(x, y) = (x + y - 4) / (√(x + y) - 2) as (x, y) approaches (2, 2) subject to the constraint x + y = 4 is 4. This result was obtained through both algebraic manipulation and the reduction to a single-variable limit, highlighting the versatility of different techniques in limit evaluation. The key takeaway is to always simplify the expression as much as possible before attempting to evaluate the limit and to be mindful of indeterminate forms, which often signal the need for further analysis.

By mastering these techniques, one can confidently tackle a wide range of limit problems in multivariable calculus, paving the way for deeper exploration of advanced mathematical concepts.