Problem 7 - Calculus Early Transcendentals

Question

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y \rightarrow(2,1)} \frac{4-x y}{x^{2}+3 y^{2}}$$

Step-by-Step Solution

Answer

The limit exists and is \( \frac{2}{7} \).

Step 1: Identify the limit expression

We are given the limit expression \( \lim_{(x,y) \to (2,1)} \frac{4 - xy}{x^2 + 3y^2} \). This means we need to evaluate the expression as \(x\) approaches 2 and \(y\) approaches 1.

Step 2: Substitute the target point into the expression

Substitute \(x = 2\) and \(y = 1\) directly into the numerator and denominator:- Numerator: \(4 - xy = 4 - 2 \times 1 = 4 - 2 = 2\)- Denominator: \(x^2 + 3y^2 = 2^2 + 3 \times 1^2 = 4 + 3 = 7\)

Step 3: Evaluate the limit value

Since both the numerator and the denominator substitute to non-zero values, the limit can be evaluated by direct substitution:\[\frac{2}{7}\]

Step 4: Conclude the limit evaluation

The direct substitution results in a finite value. Hence, the limit exists and the calculated value is \( \frac{2}{7} \).

Key Concepts

Limit Evaluation

Limit evaluation in multivariable calculus involves understanding how a function approaches a particular value as the variables within that function approach specific numbers. This is similar to single-variable limits, but it involves more than one variable, adding complexity.

To evaluate multivariable limits, you often begin by substituting the values of the variables directly into the function. However, the existence of the limit relies on the function approaching the same value regardless of the path taken as variables approach their limits.

• If a function approaches different values depending on the path taken, the limit does not exist.
• Path tests, where you check the limit along different lines or curves, can help determine if the limit may not exist.

If direct substitution results in a defined number, and you suspect that this result is consistent across all possible paths, you conclude that the limit exists.

Direct Substitution

Direct substitution is one of the simplest and most straightforward methods of finding a limit. In many cases, particularly when the function does not lead to indeterminate forms like \(\frac{0}{0}\), this method provides immediate results.

For the function \(\lim_{(x,y) \to (2,1)} \frac{4 - xy}{x^2 + 3y^2}\), we substitute the values \(x = 2\) and \(y = 1\) directly into the expression.

• Numerator calculation: \(4 - 2 \times 1 = 2\).
• Denominator calculation: \(2^2 + 3 \times 1^2 = 7\).

As neither the numerator nor the denominator results in zero, the expression evaluates to \(\frac{2}{7}\).

It's important to note that direct substitution only works smoothly when there is no division by zero or other undefined operations. It becomes a tool to quickly determine the limit when paths show consistent results.

Multivariable Calculus

Multivariable calculus builds upon single-variable calculus by exploring functions of several variables. It examines rates of change and accumulations in multidimensional spaces.

When dealing with limits in multivariable calculus:

• Variables often approach a point from many different directions or paths.
• This requires considering multiple rates of change simultaneously.
• It challenges one to consider geometric interpretations of functions.

The continuity and differentiability considerations extend beyond linear paths and involve surface and volume scaling.

In practice, multivariable calculus allows for deeper exploration of physical phenomena, such as fluid dynamics and electromagnetism, which depend on variables changing in space over time.