Problem 8 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the limit expression

We need to determine the limit of the function \( \frac{x y}{x^2 + y^2} \) as \((x, y)\) approaches \((1, 1)\).

Step 2: Substitute the limit point into the expression

Substitute \(x = 1\) and \(y = 1\) into the expression: \[\frac{1 \cdot 1}{1^2 + 1^2} = \frac{1}{1 + 1} = \frac{1}{2}\]

Step 3: Check the expression's behavior as it approaches the limit point

Since direct substitution did not result in any indeterminate form, and the expression evaluates to \( \frac{1}{2} \) without any issues, there are no complications in the limit calculation.

Key Concepts

Properties of Limits

In multivariable calculus, the properties of limits are essential tools that help us evaluate limits intuitively and accurately. These properties are rules that guide us in simplifying and breaking down complex limit expressions into parts that are easier to manage. Some of these key properties include:

• Sum and Difference Rule: The limit of the sum or difference of two functions is the sum or difference of their limits.
• Product Rule: The limit of the product of two functions is the product of their limits.
• Quotient Rule: The limit of the quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero.
• Constant Multiple Rule: The limit of a constant multiplied by a function is the constant multiplied by the limit of the function.

These properties simplify evaluating limits, especially in expressions involving multiple variables. In our given problem, these properties helped us see that we could separate the limit into manageable parts, evaluating the numerator and denominator via substitution to check the behavior near the limit point.

Approaching Limit Points

Approaching a limit point in multivariable calculus involves analyzing the behavior of a function as its variables move toward specific values. It is crucial to understand how to approach from different paths.
In the exercise, as \((x, y)\) approaches \( (1,1) \), we consider the overall behavior of the function. Since multivariable functions can be approached from various paths (linear, curved, etc.), it’s necessary to account for this by considering if the limit remains consistent from these different directions. If so, the limit is well-defined.
In this exercise, approaching directly along the path where \( x = y \) helped demonstrate consistency, as substituting these values did not lead us to an indeterminate form and directly showed us a definite result. This confirms the stability of the function as it nears the limit point.

Direct Substitution

Direct substitution is a straightforward method in calculus where we simply substitute the values of the variables into the function to find the limit.
If the function at the approaching point does not result in an indeterminate form, direct substitution can immediately give the correct limit value. In our problem, substituting \( x = 1 \) and \( y = 1 \) into the function \( \frac{x y}{x^2 + y^2} \) yields \( \frac{1}{2} \), since it resulted in a non-zero denominator value.
This method is quick and efficient for finding limits in cases where the function behaves nicely, as no further simplification or algebraic manipulation is required. It's usually the first step in limit evaluation if no immediate complexities are detected.

Indeterminate Forms

Indeterminate forms occur when substituting limit values into a function results in an uncertain or undefined expression, such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These forms do not directly provide limit values, and additional techniques are necessary to resolve them, such as L'Hôpital's rule or algebraic manipulation.
In the given problem, direct substitution did not lead to an indeterminate form, because the function gave a clear result of \(\frac{1}{2}\). However, understanding indeterminate forms is crucial as they often appear in more complex scenarios involving multivariable functions.
When faced with an indeterminate form, exploring alternative paths or employing algebraic tactics to transform the expression into a determinate form is essential to find the limit.