Problem 3 - Calculus for Biology and Medicine

Question

In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(2,-1)}\left(x^{2} y^{3}-3 x y\right) $$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Understanding the Limit Expression

The expression given is \( \lim _{(x, y) \rightarrow(2,-1)} (x^2 y^3 - 3xy) \). We need to find the limit of this expression as \((x, y)\) approaches \((2, -1)\).

Step 2: Apply the Limit Laws

The properties of limits tell us that the limit of a sum or difference is the sum or difference of the limits, and the limit of a product is the product of the limits. Hence, we can separate the expression: \[ \lim _{(x, y) \rightarrow(2,-1)} x^2 y^3 - \lim _{(x, y) \rightarrow(2,-1)} 3xy \]

Step 3: Evaluate Each Component Individually

Evaluate each component separately by plugging \(x = 2\) and \(y = -1\) into the expressions:1. For \(x^2 y^3\), substitute to find \(2^2 (-1)^3 = 4*(-1) = -4\).2. For \(3xy\), substitute to find \(3*2*(-1) = -6\).

Step 4: Combine the Results

Subtract the results from previous calculations:\[-4 - (-6) = -4 + 6 = 2\].

Key Concepts

Limits

A limit in calculus is a fundamental concept that helps us understand the behavior of functions as they approach a specific point. It essentially asks the question: "What value is a function approaching as its input gets arbitrarily close to a certain point?" Let's say we have a function \( f(x) \), and we want to find the limit as \( x \) approaches some value \( c \). What we are actually interested in is the value that \( f(x) \) is getting closer to as \( x \) gets closer to \( c \). This is written mathematically as \( \lim_{x \to c} f(x) \). In the original exercise, the limit expression involves a function of two variables, \( x \) and \( y \). The goal is to find the value the function is approaching as both variables simultaneously get closer to specified values. Understanding limits is crucial for comprehending more complex topics in calculus, such as derivatives and integrals, which fundamentally rely on the concept of approaching or "tending towards" a certain value.

Multivariable Functions

When dealing with functions of more than one variable, such as \( f(x, y) \), you are working with multivariable functions. These functions have input values that consist of several variables, making them more complex compared to functions with a single variable. The primary outcome when exploring multivariable functions is understanding how the function behaves as each of the variable inputs changes simultaneously or independently. In the example from the exercise, we have a function which depends on both \( x \) and \( y \), namely \( x^2 y^3 - 3xy \). Calculating limits of multivariable functions involves seeing how the function behaves as both variables approach specified values. Graphically, instead of looking at a curve (which is what happens with single-variable functions), we are looking at surfaces or shapes in the three-dimensional space, providing more insights into the function's behavior. Studying multivariable functions is critical in fields such as engineering and physics, where models often depend on more than one changing variable.

Limit Laws

Limit laws are foundational rules that simplify the process of finding limits. These laws state that the limit of a sum is equal to the sum of the limits, and the limit of a product is equal to the product of the limits. When we apply these laws in the context of the exercise, we separate the expression into different parts to make calculations simpler:

• The expression \( \lim_{(x, y) \to (2,-1)} x^2 y^3 - \lim_{(x, y) \to (2,-1)} 3xy \) utilizes these limit laws by breaking down the components.
• This allows us to evaluate each part independently by directly substituting \( x = 2 \) and \( y = -1 \) into them to get their limit.

These rules make it much more straightforward to solve complex limit problems by transforming them into simpler, more manageable pieces. Familiarity with limit laws is essential because they bridge basic algebraic manipulations to the more abstract concepts of calculus.

Evaluation of Limits

Evaluating limits, especially in multivariable calculus, can seem daunting at first because it involves understanding the behavior of a function as its inputs change together. To evaluate limits, we often follow these steps:

• First, understand the expression and identify the inputs involved.
• Use limit laws to break down complex expressions into simpler parts wherever possible.
• Substitute the limits of each variable independently and calculate the result for each part.
• Combine the results from the simpler parts to find the overall limit.

In our exercise, we first isolated each component of the multivariable expression using limit laws and then substituted specific values to evaluate each component individually. For example:

• Substituting \( x = 2 \) and \( y = -1 \) into \( x^2 y^3 \) gives us \( -4 \).
• Into \( 3xy \), it gives \( -6 \).

Finally, the results are combined to find the limit of the entire expression, which in this case turned out to be \( 2 \). This structured approach not only makes evaluating limits manageable but also builds an intuitive understanding of how functions behave as their inputs change.