Problem 12 - Calculus for Biology and Medicine

Question

In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(-1,-2)} \frac{x^{2}-y^{2}}{2 x y+2} $$

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{2}\).

Step 1: Identify the Limit

The given limit is \( \lim _{(x, y) \rightarrow(-1,-2)} \frac{x^{2}-y^{2}}{2 x y+2} \). This means you need to find the value of the expression as \(x\) approaches \(-1\) and \(y\) approaches \(-2\).

Step 2: Substitute Limit Values

Substitute \(x = -1\) and \(y = -2\) directly into the expression \( \frac{x^{2}-y^{2}}{2 x y+2} \). This gives \( \frac{(-1)^{2} - (-2)^{2}}{2(-1)(-2) + 2} \).

Step 3: Calculate the Numerator

Calculate the numerator \((-1)^{2} - (-2)^{2}\). First, square \(-1\) to get \(1\) and \(-2\) to get \(4\), then subtract: \(1 - 4 = -3\).

Step 4: Calculate the Denominator

Calculate the denominator \(2(-1)(-2) + 2\). Multiply \(2 \times -1 \times -2 = 4\), then add \(2\): \(4 + 2 = 6\).

Step 5: Calculate the Limit

Now, divide the results from the numerator and the denominator: \( \frac{-3}{6} = \frac{-1}{2} \).

Key Concepts

Understanding the Properties of Limits

Limits are foundational in calculus and allow us to investigate the behavior of functions as the input approaches a certain value. The properties of limits simplify the process of calculating limits and provide tools that make the work easier.

• **Linearity**: If \(\lim_{x \to a} f(x) = L\) and \(\lim_{x \to a} g(x) = M\), then \(\lim_{x \to a} [f(x) + g(x)] = L + M\).
• **Scalar Multiplication**: \(\lim_{x \to a} [c \cdot f(x)] = c \cdot L\) where \(c\) is a constant.
• **Product**: The limit of a product is the product of the limits: \(\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M\).
• **Quotient**: The limit of a quotient is the quotient of the limits: \(\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}\), provided \(M eq 0\).

Applying these properties allows for direct substitution when the expression is continuous at the point of interest. In exercises involving complex expressions, breaking these down into simpler components using the properties of limits aids in solving parts independently before combining results. This step-by-step process is useful as seen in exercises such as the provided limit problem.

Introduction to Multivariable Functions

Calculating limits for multivariable functions, such as \( rac{x^{2}-y^{2}}{2xy+2}\), explores functions with more than one variable. Unlike single-variable functions that handle one input at a time, multivariable functions deal with several inputs that move independently.
This poses different challenges compared to single-variable functions, primarily because the limit could potentially change depending on the path chosen to approach the point.
Therefore, to ensure the existence of a limit, the function must approach the same value regardless of the path taken towards the point. This involves not just straightforward calculations but understanding the geometric and analytic behavior of the function in multiple dimensions.

• Understanding that limits must be the same from all directions signifies maintaining consistency across different dimensions.
• Transforming multivariable functions into simpler forms helps visualize and understand their behavior.
• Substituting specific values into the function, as shown in the original exercise, provides clarity on predicted outputs as variables approach specific points.

Techniques for Limit Evaluation

Limit evaluation involves working out precisely what a function's value is heading toward as we approach some point. It's crucial in understanding and predicting the behavior of systems described by these functions.
There are several methods commonly used:

• **Substitution Method**: Often the simplest method, replacing variables with their limits directly when the expression is continuous or appears to be straightforward.
• **Factoring and Simplification**: In cases where direct substitution leads to an indeterminate form like 0/0, factoring or simplifying the expression may resolve issues and facilitate finding the limit.
• **Rationalization**: Could be used if the expression involves roots; multiplying and dividing by the conjugate may aid in the process.

In the original exercise, substitution is employed directly after ensuring there are no indeterminate forms present, simplifying the process and ensuring a quick evaluation of the limit. This reflects the practicality of choosing the right technique to fit the given problem constraints effectively.