Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(-2,1)}\left(x y^{3}-x y+3 y^{2}\right)\)

Step-by-Step Solution

Answer

The limit is 3.

Step 1: Understand the Problem Statement

We are given a function expressed as \( f(x, y) = xy^3 - xy + 3y^2 \) and we need to evaluate the limit as \( (x, y) \) approaches \((-2, 1)\). This task involves checking if the limit exists at a specific point in a two-variable function, which means the value should be the same from all paths of approach.

Step 2: Substitute the Point into the Function

Since the function \( f(x, y) \) is defined and continuous around \( (x, y) = (-2, 1) \), we can substitute \( x = -2 \) and \( y = 1 \) directly into the function to find the limit.Let's compute:\[f(-2, 1) = (-2)(1)^3 - (-2)(1) + 3(1)^2\]

Step 3: Calculate Each Term Separately

Evaluate each term in the expression:1. \( (-2)(1)^3 = -2 \)2. \( -(-2)(1) = 2 \)3. \( 3(1)^2 = 3 \)

Step 4: Simplify the Expression

Combine the calculated terms from the previous step:\[ -2 + 2 + 3 \]

Step 5: Final Calculation

Add the terms together:\[ -2 + 2 + 3 = 3 \]

Step 6: Conclusion of Existence of the Limit

Since we were able to substitute the point into the function without any undefined terms and got a finite answer, the limit does exist. Therefore, the limit of \( f(x, y) \) as \( (x, y) \rightarrow (-2, 1) \) is 3.

Key Concepts

Continuous Functions

In mathematics, a function is continuous if, roughly speaking, you can draw it without lifting your pen from the paper. A continuous function does not have any breaks, holes, or jumps.
For multivariable functions like the one given, continuity means something similar: small changes in input lead to small changes in output.
This continuity is key when evaluating limits because it allows for direct substitution, as seen in the original problem. When a function is continuous at a point, like the point ext{(-2, 1)} in the problem, the limit can simply be evaluated by plugging the coordinates into the function.
It's important to note that while continuity simplifies limit evaluations, not all functions are continuous everywhere. However, for polynomial functions or rational functions with non-zero denominators at the point of interest, continuity is maintained.

Multivariable Calculus

Multivariable calculus extends the ideas of calculus to functions of several variables. In contrast to single-variable calculus, where there's only one input, multivariable calculus deals with functions of two or more variables.
These functions can be represented as surfaces in three-dimensional space and involve a more complex analysis in order to locate things like maxima, minima, and limits.
With multivariable functions, evaluating limits becomes slightly more challenging because inputs can approach from any direction in a plane, making the path of approach crucial. Multivariable calculus provides the tools to work with these complex functions, investigate their behavior, and understand how changes in every variable affect the whole system.

Limit Evaluation

Limit evaluation is the process of finding the value that a function approaches as the input approaches a particular point. In multivariable functions, it refers to finding what value the function heads towards as two or more variables approach some fixed point.
In our exercise, the limit evaluation process becomes easier due to the function's continuity. We can substitute directly the values into the equation since all the terms are determinants and no division by zero occurs.

• Find and clearly substitute the inputs into the function.
• Simplify term by term.
• Combine the results to determine the overall value.

This evaluation confirms whether a function has a valid limit at a particular point or whether it varies based on the direction of approach.

Mathematical Analysis

Mathematical analysis involves breaking down functions and sequences to understand their behavior. Analyzing limits like in the exercise allows for the deeper understanding of a function's behavior nearby certain points.
Here, we determined that the function approaches 3 as ext{(x, y)} approaches ext{(-2, 1)} by calculating the value directly, which provides both an answer and insight into the stability of the function near that point.
Through this analysis, important properties of functions, such as continuity, differentiability, and integrability, can be investigated, allowing for predictions about their graph and behavior in general terms.
This is essential for a broad scope of sciences such as physics, engineering, and economics, where modeling and understanding change is crucial.