Problem 17 - The Calculus with Analytic Geometry

Question

In Exercises 17 through 19, evaluate the given limit by the use of limit theorems. \(\lim _{(x, y) \rightarrow(2,3)}\left(3 x^{2}+x y-2 y^{2}\right)\)

Step-by-Step Solution

Answer

The limit is 0.

Step 1 - Understand the limit

To evaluate the limit \(\lim _{(x, y) \rightarrow (2,3)}(3x^{2} + xy - 2y^{2})\), approach the given function as \(x\) approaches 2 and \(y\) approaches 3.

Step 2 - Substitute the values

Substitute \(x = 2\) and \(y = 3\) directly into the function \(3x^2 + xy - 2y^2\).

Step 3 - Evaluate the expression

Calculate the function step-by-step: - Compute \(3(2)^2\) which equals 12 - Compute \(2 \times 3\) which equals 6 - Compute \(-2(3)^2\) which equals -18

Step 4 - Sum the results

Combine the results from the previous step: \( 12 + 6 - 18 = 0 \)

Key Concepts

Evaluating Limits

Evaluating limits is an essential concept in calculus. It helps us understand the behavior of functions as they approach specific points. For single-variable functions, the limit describes what value the function approaches as the variable gets closer to a certain number.
In multivariable calculus, we need to consider limits as all variables approach specific points. In this exercise, we evaluated \(\lim _{(x, y) \rightarrow(2,3)}(3x^{2}+xy-2y^{2})\). This involves:

• Approaching the function by substituting the values of x and y into the function
• Simplifying and evaluating the expression at those points

In our example, substituting x = 2 and y = 3 directly into the function 3x^2 + xy - 2y^2 allowed us to find the limit.
Remember, a limit exists if the function approaches a single, specific value as the variables approach their points.

Limit Theorems

Limit theorems are essential tools in calculus for finding limits more easily. They provide rules that help us simplify and evaluate limits. Some key limit theorems include:

• Sum Rule: If the limits of two functions exist, the limit of their sum is the sum of their limits.
• Product Rule: If the limits of two functions exist, the limit of their product is the product of their limits.
• Quotient Rule: If the limits of two functions exist, and the limit of the denominator is not zero, the limit of their quotient is the quotient of their limits.
• Constant Multiple Rule: The limit of a constant times a function is the constant times the limit of the function.

In our example, applying these rules helps streamline the process of evaluating the limit. Substituting and then calculating each term separately aligns with these rules.
By understanding and applying these theorems, you can simplify complex limit problems and evaluate them more effectively.

Multivariable Functions

Multivariable functions involve more than one input variable, such as f(x, y). When working with these functions, you must consider how changes in one variable affect the overall function.
For example, in the function \(\lim _{(x, y) \rightarrow(2,3)}(3x^{2}+xy-2y^{2})\), the variables x and y independently contribute to the function's value. Evaluating limits for these functions requires:

• Analyzing the function as each variable approaches its specific point.
• Combining the results logically to understand how the function behaves near that point.

In multivariable calculus, evaluating limits can be more challenging due to the added complexity of multiple variables. However, by breaking down the problem step-by-step, as demonstrated in our example, you can efficiently find the limit.
Always consider the interaction between variables and utilize limit theorems to aid in your calculations.