Problem 13 - Calculus for Biology and Medicine

Question

In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(2,0)} \frac{2 x+4 y^{2}}{y^{2}+3 x} $$

Step-by-Step Solution

Answer

The limit is \( \frac{2}{3} \).

Step 1: Substitute Directly into the Expression

To find \( \lim_{(x, y) \rightarrow (2, 0)} \frac{2x + 4y^2}{y^2 + 3x} \), start by substituting \(x = 2\) and \(y = 0\) directly into the expression: \( \frac{2(2) + 4(0)^2}{(0)^2 + 3(2)} = \frac{4 + 0}{0 + 6} = \frac{4}{6} \). This simplifies to \( \frac{2}{3} \).

Step 2: Conclude the Limit

Since substituting directly gives a valid numeric result, the limit exists and is correct. The evaluation shows that \( \lim_{(x, y) \rightarrow (2, 0)} \frac{2x + 4y^2}{y^2 + 3x} = \frac{2}{3} \).

Key Concepts

Understanding Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus into higher dimensions. It involves functions of two or more variables like \( f(x, y) \) or \( f(x, y, z) \). These functions are explored to understand how they behave as their inputs change. This is especially useful in fields like physics and engineering where you have to consider multiple factors simultaneously.
In multivariable calculus, limits help us find the behavior of a function as the variables approach a certain point. This is not always straightforward due to the added complexity of multiple paths that variables can take to approach a point. Understanding these limits is crucial for finding derivatives and integrals of functions with multiple variables. These concepts allow us to examine how systems change over time or space, leading to insights in dynamic environments.

Direct Substitution in Calculus

Direct substitution is one of the simplest methods to evaluate limits. Whenever a function is continuous at a point, you can find the limit by directly substituting the values of the variables into the function. This process helps determine the value that the function approaches as the variables get infinitely close to a target point.
In our exercise, the direct substitution process was straightforward:

• Substitute \( x = 2 \) and \( y = 0 \) into the function \( \frac{2x + 4y^2}{y^2 + 3x} \).
• We calculated the expression to get \( \frac{2}{3} \), showing that the limit exists and is finite.

Direct substitution only works when the function does not lead to undefined values, like division by zero, after substitution. Hence, checking for continuity is a necessary step before making direct substitutions.

Limit Properties for Multivariable Functions

Limit properties help simplify calculations and understand behaviors of functions better. The properties of limits used in multivariable calculus are extensions of those used in single-variable calculus. Here are some useful limit properties:

• Sum/Difference Rule: The limit of a sum/difference is the sum/difference of the limits: \( \lim_{(x, y) \rightarrow (a, b)} [ f(x, y) \pm g(x, y) ] = \lim_{(x, y) \rightarrow (a, b)} f(x, y) \pm \lim_{(x, y) \rightarrow (a, b)} g(x, y) \).
• Product Rule: The limit of a product is the product of the limits: \( \lim_{(x, y) \rightarrow (a, b)} [ f(x, y) \times g(x, y) ] = \lim_{(x, y) \rightarrow (a, b)} f(x, y) \times \lim_{(x, y) \rightarrow (a, b)} g(x, y) \).
• Quotient Rule: The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero: \( \lim_{(x, y) \rightarrow (a, b)} \frac{f(x, y)}{g(x, y)} = \frac{\lim_{(x, y) \rightarrow (a, b)} f(x, y)}{\lim_{(x, y) \rightarrow (a, b)} g(x, y)} \).

In our problem, we directly applied the quotient rule because the denominator did not approach zero. Understanding these properties allows for effective evaluation of limits, ensuring correct solutions.