Problem 6 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The limit is \(e\).

Step 1: Understanding the Problem

We need to find the limit of the function \(f(x,y) = e^{-xy} \cos(x+y)\) as \((x,y)\) approaches \((1,-1)\).

Step 2: Substituting the Values

Substitute \(x = 1\) and \(y = -1\) directly into the function: \(f(x,y) = e^{-1(-1)} \cos(1 + (-1)) = e^1 \cos(0)\).

Step 3: Simplifying Exponential and Trigonometric Terms

Since \(e^1 = e\) and \(\cos(0) = 1\), the function simplifies to \(e^1 \times 1 = e\).

Step 4: Calculating the Limit

As \((x,y)\) approaches \((1,-1)\), the limit of the function \(f(x,y)\) is \(e\) because substituting the values directly gives a valid result with no indeterminate form or discontinuities.

Key Concepts

Limits in Multivariable Calculus

Limits in multivariable calculus are similar to limits in single-variable calculus, but they involve functions with more than one variable. Multivariable limits explore how a function behaves as its input values approach a particular point from all directions in its domain.
This is more complex than single-variable limits because we need to consider different paths of approach, as the outcome can vary based on the path taken.

• A limit in two variables, \((x, y)\) approaching \(a, b\), is defined only if the result is the same regardless of the path taken to \(a, b\).


• If you get different results using different paths, the limit does not exist.

Understanding these concepts helps us determine continuity and differentiability of multivariable functions.

Evaluating Limits

Evaluating limits in multivariable calculus often involves substituting the approaching point into the function, if it's continuous at that point. If substitution leads directly to a finite number, then that's the limit.
When direct substitution does not work due to indeterminate forms, other methods are used.

• Methods like path approaching might be used, where we evaluate the limit using several different paths to check if results remain consistent.


• If the paths give different results, the limit does not exist.
• Simplifying the function using algebraic manipulation can often resolve indeterminate forms.

In our original exercise, substitution directly provided a finite and consistent result, making our task straightforward.

Exponential Functions

Exponential functions, commonly expressed as \(e^x\), are vital in calculus due to their unique properties. These functions are continuously growing and are defined for any real number.
Characteristics include constant percentage growth, which models numerous natural phenomena.

• The derivative of \(e^x\) is itself, \(e^x\), which simplifies many calculus problems.
• In multivariable calculus, they behave similarly, although they'll often appear as part of more complex expressions.

In the problem on hand, \(e^{-xy}\) forms part of the expression, and we needed its evaluation to determine the entire function's limit as \((x, y)\) approached \((1, -1)\). When calculated, it simplified to a core term, facilitating solution.

Trigonometric Functions

Trigonometric functions like \(\cos(x)\) are integral in calculus for analyzing periodic phenomena. These functions repeat every \(2\pi\) units, providing essential tools for understanding waves and oscillations.
Using identities and properties of these functions aids in simplifying calculus calculations.

• The cosine function ranges from -1 to 1, making it bounded and predictable.
• When evaluating a limit involving a trigonometric function, using known values at specific points, such as \(\cos(0) = 1\), can quickly simplify calculations.

In our exercise, \(\cos(x+y)\) evaluated directly to \(\cos(0) = 1\), eliminating complexity and allowing an easy calculation of the limit.