Problem 9 - Calculus for Biology and Medicine

Question

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(2.1)} \frac{x^{2}+y^{2}}{x^{2}-y^{2}}\)

Step-by-Step Solution

Answer

The limit is \(\frac{5}{3}\).

Step 1: Identify the Type of Limit

We have a multivariable limit \[ \lim_{(x, y) \rightarrow (2,1)} \frac{x^2 + y^2}{x^2 - y^2} \]We need to evaluate this limit as \((x, y)\) approaches \((2, 1)\).

Step 2: Simplifying the Function

First, plug in the values \(x = 2\) and \(y = 1\) into the expression \(\frac{x^2 + y^2}{x^2 - y^2}\) to see if it's defined:\[\frac{2^2 + 1^2}{2^2 - 1^2} = \frac{4 + 1}{4 - 1} = \frac{5}{3}\] As the expression does not result in an indeterminate form like \(\frac{0}{0}\), the limit can be directly evaluated.

Step 3: Evaluate the Limit

Since substituting \((x, y) = (2, 1)\) into the function \(\frac{x^2 + y^2}{x^2 - y^2}\) yields a defined value without indeterminacy, the limit is the value of the function at that point:\[\lim _{(x, y) \rightarrow(2, 1)} \frac{x^{2}+y^{2}}{x^{2}-y^{2}} = \frac{5}{3}\]

Key Concepts

Calculus

Calculus is a branch of mathematics that enables us to understand the change and motion through the use of derivatives and integrals. One important aspect of calculus is the concept of limits. Limits help us to understand the behavior of a function as it approaches a certain point. This is crucial in determining continuity, finding tangents, and evaluating multivariable functions, like in our exercise where we approach \\((x, y) \rightarrow (2,1)\).

• Limits are fundamental in calculus for analyzing functions.
• They allow us to see what happens as inputs approach a certain point.
• Understanding limits is essential for mastering calculus and for tackling multivariable problems.

To solve calculus problems effectively, it's important to grasp the intuitive idea of a limit, which is what a function approaches as it gets closer to a specific input value. This understanding is foundational for tackling more complex calculus concepts like derivatives and integrals.

Properties of Limits

Understanding the properties of limits is key to evaluating the limits of complex functions, especially when dealing with multivariable calculus. These properties provide rules that help us break down and simplify problems to find limit values more easily.Some basic properties that are often used include:

• Sum Rule: The limit of a sum is the sum of the limits.
• Difference Rule: The limit of a difference is the difference of the limits.
• Product Rule: The limit of a product is the product of the limits.
• Quotient Rule: The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero.

In our example, the use of these properties allowed the direct substitution of \(x = 2\) and \(y = 1\) to find that the expression \( \frac{x^{2} + y^{2}}{x^{2} - y^{2}} \) simplifies to \( \frac{5}{3} \). This substitution worked directly because it didn't result in an indeterminate form.

Indeterminate Forms

Indeterminate forms occur when the limits of functions result in undefined expressions such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These forms require special techniques to evaluate, often involving algebraic manipulation or L'Hopital's Rule if applicable.In the exercise, when substituting \( x = 2 \) and \( y = 1 \), the expression \( \frac{x^{2} + y^{2}}{x^{2} - y^{2}} \) did not result in an indeterminate form. This means it evaluated smoothly to \( \frac{5}{3} \). When faced with an indeterminate form, you might consider:

• Simplification: Algebraically simplify the function, if possible.
• L'Hopital's Rule: Apply this rule if both numerator and denominator tend to zero or infinity.
• Rewriting expressions: Use trigonometric identities, factoring, or common factors to simplify.

Handling indeterminate forms often involves creative problem-solving and a thorough understanding of limit properties. However, when they are not present, as in this case, the solution can be reached by direct evaluation.