Problem 19 - Calculus Early Transcendentals

Question

5-22 Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y, z) \rightarrow(3,0,1)} e^{-x y} \sin (\pi z / 2)$$

Step-by-Step Solution

Answer

The limit exists and is 1.

Step 1 - Identify the function

The function to analyze is given as \( e^{-xy} \sin(\pi z / 2) \). We need to consider what happens to this function as \((x, y, z)\) approaches \((3, 0, 1)\).

Step 2 - Substitute the limit point

Substitute \( x = 3 \), \( y = 0 \), and \( z = 1 \) into the function. Evaluate the expression: \[ \lim_{(x, y, z) \rightarrow (3, 0, 1)} e^{-xy} \sin(\pi z / 2) = e^{-3 \cdot 0 } \sin(\pi \cdot 1 / 2). \]

Step 3 - Simplify the expression

Calculate each part separately:- Since \( xy = 3 \cdot 0 = 0 \), then \( e^{-xy} = e^0 = 1 \).- Next, evaluate \( \sin(\pi / 2) \). Since \(\sin(\pi / 2) = 1\), we conclude the expression simplifies as follows:\[ e^0 \cdot \sin(\pi / 2) = 1 \cdot 1 = 1. \]

Step 4 - Conclude the limit

From the simplifications, we see that the limit is a constant value. Thus, the limit exists and is equal to \(1\).

Key Concepts

Limits in Three Dimensions

When dealing with multivariable calculus, it's crucial to understand limits in three dimensions. In this context, we're handling functions with variables that approach specific points in a 3D space. Picture a function dependent on three variables:

• **(x, y, z)** represents the variables in our space.
• **(3, 0, 1)** is the point we're moving towards.

Imagine moving closer to \( (3, 0, 1) \) along different paths in a 3D grid. The limit checks if your function's value approaches a single, definite number regardless of the path taken. If so, the 3D limit exists. Here, substituting values directly into the function helps determine the limit value.To ensure the limit exists, consider:

• The result should be independent of the path taken to approach the limit point.
• Multiple paths should yield the same endpoint value.

Exponential Functions

Exponential functions are foundational in calculus, particularly because of their powerful growth or decay behavior. In our exercise, the function part to focus on is \( e^{-xy} \).

• The base, \( e \), is approximately 2.718, a constant known as Euler's number.
• The exponent is \( -xy \), affecting how the function behaves.

For the given problem, when \( x = 3 \) and \( y = 0 \), the expression simplifies greatly:

• The product \( xy = 0 \), turning \( e^{-xy} \) into \( e^0 \).
• Any non-negative number raised to the zero power equals 1, thus \( e^0 = 1 \).

Understanding this simplification is critical as it showcases how exponential functions can stabilize to a constant value under specific conditions.

Trigonometric Functions

Trigonometric functions, like the sine function, play a critical role in calculus and many applied fields. Here, we need to evaluate \( \sin(\pi z / 2) \).

• Sine functions exhibit periodic behavior, cycling between -1 and 1.
• In our case, \( z = 1 \), making the calculation straightforward: \( \pi \cdot 1 / 2 = \pi / 2 \).
• The value of \( \sin(\pi / 2) \) equals 1.

This aspect simplifies the limit calculation, as the overall function value heavily depends on this trigonometric evaluation. Once simplified, the sine function provides a consistent value, allowing us to confirm the limit's existence in the problem.

Limit Evaluation Techniques

Evaluating limits, particularly in three dimensions, involves several strategies that determine whether a limit exists. This task often requires multiple approaches or techniques to confirm results:

• **Substitution:** Directly substituting values into the function, as done here, can immediately resolve simpler functions.
• **Path Approach:** Checking limits along different paths can also ensure consistency.

In the exercise, substituting \( (3, 0, 1) \) directly provided us the limit.

• Because both the exponential and trigonometric components simplified to 1, the overall limit equaled 1, showing a straightforward scenario.
• If substitution doesn't work, paths might include approaching from axis-aligned directions or diagonally across the coordinate planes.

Limit evaluation techniques thus streamline the process, confirming whether a 3D function's limit is consistent and the same from any approach.