Problem 2 - Calculus with Concepts in Calculus

Question

Evaluate the limit. $$ \lim _{(x, y) \rightarrow(1,-2)}\left(2 x^{3}-4 x y+5 y^{2}\right) $$

Step-by-Step Solution

Answer

The limit is 30.

Step 1: Understand the Problem

We are given a multivariable limit problem. Our task is to find the limit of the function \( f(x, y) = 2x^3 - 4xy + 5y^2 \) as the point \((x, y)\) approaches \((1, -2)\).

Step 2: Substitute Directly

First, let's substitute \(x = 1\) and \(y = -2\) directly into the function to evaluate the limit:\[f(x, y) = 2(1)^3 - 4(1)(-2) + 5(-2)^2\]

Step 3: Calculate the Expression

Calculate each part of the expression one by one:- \(2(1)^3 = 2\)- \(-4(1)(-2) = 8\)- \(5(-2)^2 = 20\)Combine these results:\[2 + 8 + 20 = 30\]

Step 4: Conclusion

Since the function is made of polynomials, it's continuous everywhere. Thus, direct substitution gives the correct result when taking the limit. Therefore, the limit is \(30\).

Key Concepts

Continuous Functions

Continuous functions are fundamental in calculus because they allow us to evaluate limits straightforwardly by plugging in values, as they don’t have breaks or gaps. Polynomials, as we have in our example with terms like \(2x^3\) or \(4xy\), are the most straightforward type of continuous functions due to their smooth curves.Here are some important points about continuous functions:

• They have no interruptions. This means functions can be drawn without lifting the pen from the paper.
• For functions involving polynomials, continuity is guaranteed over all real numbers.
• Continuous functions mean limits can be found by direct substitution, making them very handy in calculating multivariable limits.

In our solved exercise, the polynomial function given is continuous at the point \((1, -2)\), simplifying our limit calculation by allowing direct substitution.

Direct Substitution

Direct substitution is a method of finding limits where we directly plug in the values into the function. This technique is reliable when dealing with continuous functions like polynomials.

In our problem with the given multivariable function, direct substitution was applied as follows:

• The process involved taking the point \((1, -2)\) and substituting it into the function \(f(x, y) = 2x^3 - 4xy + 5y^2\).
• Each component of the expression was calculated with these values.

For continuous functions, direct substitution is seamless because the limit of a function at a point is the same as the value of the function at that point.For any function that is complicated, non-continuous, or involves division by zero, direct substitution may not be appropriate. In those cases, other techniques like factoring, rationalizing, or using L'Hôpital's Rule might be necessary.

Polynomials

Polynomials such as \(2x^3 - 4xy + 5y^2\) are algebraic expressions made up of variables raised to whole number powers, combined using addition, subtraction, and multiplication. These characteristics make them remarkably versatile in mathematics.Here are several aspects of polynomials:

• They are differentiable and integrable, providing flexibility in calculus.
• Since they are continuous for all real numbers, we don't need to worry about gaps or discontinuities.
• In the context of limits, they simplify the task because their limits can be assessed through direct substitution.

In our problem, the polynomial's continuity confirmed that by directly substituting the values, we computed the multivariable limit correctly. Such stability in behavior makes polynomials an invaluable tool in understanding and evaluating limits.