Problem 3 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is \(-2\).

Step 1: Identify the Function

The problem asks for the limit of the function \( f(x, y) = x^{2} y^{3} - x y \) as \((x, y)\) approaches \((2, -1)\). We'll use properties of limits to solve it.

Step 2: Apply Limit to Each Term

Split the limit across the subtraction: \[ \lim_{(x, y) \rightarrow (2, -1)} (x^{2} y^{3}) - \lim_{(x, y) \rightarrow (2, -1)} (x y) \] Using the property that limits distribute across addition and subtraction.

Step 3: Compute Each Limit

First, compute \( \lim_{(x, y) \rightarrow (2, -1)} x^{2} y^{3} \):- Substitute \(x = 2\) and \(y = -1\): - \((2)^{2} \cdot (-1)^{3} = 4 \cdot (-1) = -4\)Second, compute \( \lim_{(x, y) \rightarrow (2, -1)} x y \):- Substitute \(x = 2\) and \(y = -1\): - \(2 \cdot (-1) = -2\)

Step 4: Substitute Calculated Limits

Substitute the calculated limits into the expression: \(-4 - (-2)\).

Step 5: Simplify the Expression

Simplify the expression obtained in the previous step: \(-4 + 2 = -2\).

Key Concepts

Multivariable Functions

When we talk about multivariable functions, we are referring to functions that depend on more than one variable. For example, in the given exercise, we have the function \( f(x, y) = x^{2}y^{3} - xy \). Here, \( f(x, y) \) takes two inputs, \( x \) and \( y \), and gives a single output. Similar to single-variable functions, multivariable functions can be used to model processes that depend on multiple factors like temperature and pressure, or horizontal and vertical positions.

One of the interesting aspects of multivariable functions is their graphical representation. Instead of a simple curve on a plane, their graph forms a surface in three-dimensional space. This makes visualization a bit more challenging but also fascinating as it involves more complex shapes that can reveal rich behaviors of the function depending on how inputs are varied.

In the context of limits, it's essential to remember that for multivariable functions, calculating the limit as a point approaches a certain value involves all the variables in the function getting closer to their respective target values simultaneously.

Properties of Limits

The properties of limits are powerful tools in calculus, and they help simplify complex problems, especially in the context of multivariable calculus. In the exercise solution, one such key property is used: the distribution of limits across addition and subtraction. This property allows us to split the limit of a sum or a difference into the sum or difference of limits. For instance, \( \lim_{(x, y) \rightarrow (2, -1)}(x^2y^3 - xy) \) can be separated as individual limits:

• \( \lim_{(x, y) \rightarrow (2, -1)}(x^2y^3) \)
• \( \lim_{(x, y) \rightarrow (2, -1)}(xy) \)

This step is crucial because it enables us to handle each term separately, making the calculations more manageable.

Properties like the limit of a product being the product of the limits are extension rules that stem from basic limit properties studied in single-variable calculus. These properties ensure that if the limit of each individual function exists as separate entities, their combination will also have a well-defined limit. As you encounter more complex functions, remembering these properties will guide you toward easier solutions.

Limit Substitution

Limit substitution is a straightforward technique where, after applying the relevant properties of limits, the variables in the function are directly replaced with the specific values they approach. In the original problem, this meant substituting \( x = 2 \) and \( y = -1 \) into the separated limit expressions.

For example, to compute \( \lim_{(x, y) \rightarrow (2, -1)} x^{2} y^{3} \), we substitute the values into the expression to find:

• \( (2)^{2} \cdot (-1)^{3} \)
• which simplifies to \( 4 \cdot (-1) = -4 \)

Similarly, for \( \lim_{(x, y) \rightarrow (2, -1)} xy \), substituting gives:

• \( 2 \cdot (-1) = -2 \)

This direct substitution, made possible by solving the individual limits first, ultimately helps in determining the limit of the entire function by evaluating these simple arithmetic expressions. This method hinges on the continuity of the function and the existence of the limit, allowing us to use simple calculations to reach a solution.