Problem 10 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is 4.

Step 1: Substitute the values into the expression

First, substitute the given point \((x, y) = (-1, 3)\) into the expression \(\frac{x^2 - xy}{2x + y}\). This gives us:\[\frac{(-1)^2 - (-1)(3)}{2(-1) + 3}\]

Step 2: Simplify the numerator

Compute the numerator by calculating each term. \((-1)^2 = 1\) and \(-(-1)(3) = 3\). Thus, the numerator becomes:\[1 + 3 = 4\]

Step 3: Simplify the denominator

Compute the denominator by multiplying and adding the given values. \(2(-1) = -2\) and adding \(3\) gives:\[-2 + 3 = 1\]

Step 4: Divide the simplified terms

Take the simplified numerator and denominator, and perform the division:\[\frac{4}{1} = 4\]

Step 5: Conclude the limit

As there are no indeterminate forms or need for further simplification, conclude that the limit exists and the calculated value is:\[\lim _{(x, y) \rightarrow(-1,3)} \frac{x^2 - xy}{2x + y} = 4\]

Key Concepts

Properties of Limits

Limits in calculus are essential for understanding how functions behave as they approach a certain point. To simplify the evaluation process, we use properties of limits that allow us to break down complex expressions. These properties include:

• Linearity: The limit of a sum is the sum of the limits. Similarly, the limit of a difference is the difference of the limits.
• Scalar Multiplication: A constant can be factored out of the limit process.
• Product Rule: The limit of a product is the product of the limits, as long as each individual limit exists.
• Quotient Rule: The limit of a quotient is the quotient of the limits, provided the denominator's limit isn’t zero.
• Power Rule: The limit of a function raised to a power is the limit of the function raised to that power.

By applying these rules, we transform complex functions into simpler ones before calculating their limits. This process allows us to make safe assumptions about continuity, helping us understand function behavior at specific points without actually reaching those points.

Multivariable Calculus

Multivariable calculus deals with functions that depend on more than one variable. This isn't too different from single-variable calculus, but the scale extends across higher dimensions. Imagine we have a surface instead of a curve. Instead of looking at \( f(x) \), we now have \( f(x, y) \) or even \( f(x, y, z) \).In multivariable calculus, limits play a key role as they dictate how these functions behave when several inputs are approaching specific values simultaneously. To evaluate such limits, it’s essential to check the value from different paths because the limit only exists if it approaches the same value along all paths.For example, with \( \lim_{(x,y) \to (-1,3)} \frac{x^2 - xy}{2x + y} \), we substitute values directly in this case, but if it had resulted in an indeterminate form, further analysis would involve checking paths like \( y = x + c \) or conversion to polar coordinates for more insight.

Limit Evaluation Steps

Calculating limits, especially in complex functions, requires a methodical approach. Here’s a concise breakdown of steps you should follow:

• Substitution: The first step is always to substitute the point into the function directly. If the result isn’t an indeterminate form (like 0/0), you have your limit.
• Simplification: If substitution leads to an indeterminate form, simplify the expression. Factorization, rationalization, or other algebraic manipulation might be needed.
• Alternative Paths: Especially in multivariable calculus, check directional paths to ensure the limit is consistent from all angles.
• Numerical Approximation: When manual calculations remain tricky, graphing or numerical tools can approximate the limit quickly.

Following these steps ensures a comprehensive analysis of any given limit problem, paving the way for a more profound understanding of the function’s behavior around specific points. Over time, these approaches become second nature and vastly improve problem-solving efficiency.