Problem 13 - Calculus for Biology and Medicine

Question

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: Check for Direct Substitution

Examine the limit expression to see if direct substitution is possible. Substitute the values of \( x = 2 \) and \( y = 0 \) into the expression. You get: \( \frac{2(2) + 4(0)^2}{0^2 + 3(2)} = \frac{4}{6} \).

Step 2: Simplify the Result

Simplify the fraction obtained from substituting the values. \( \frac{4}{6} = \frac{2}{3} \). This shows that the limit can be evaluated directly through substitution and simplifies to \( \frac{2}{3} \).

Key Concepts

Direct Substitution

Direct substitution is one of the easiest methods to evaluate limits. It involves directly plugging the values into the function to see if it results in a clear answer. This method works well when both the numerator and denominator of the expression are defined and non-zero at the given point.

In our example, after substituting the values, we obtained:

• Numerator: Substitute value of \( x = 2 \) into \( 2x + 4y^2 \), resulting in \( 2(2) + 4(0)^2 = 4 \).
• Denominator: Substitute value of \( x = 2 \) and \( y = 0 \) into \( y^2 + 3x \), resulting in \( 0^2 + 3(2) = 6 \).

Now the provided expression is \( \frac{4}{6} \).

One crucial aspect of direct substitution is recognizing when it's not initially feasible, such as when a substitution leads to a \( \frac{0}{0} \) form or an undefined expression. In such cases, further steps, like simplifying or using other limit-solving techniques, might be required. However, our expression was well-defined, allowing for a straightforward substitution.

Fraction Simplification

Fraction simplification is an essential skill in mathematics that helps in reducing expressions to their simplest forms. This process makes calculations easier to handle and final results clearer. After performing direct substitution in our problem, the expression we were left with was \( \frac{4}{6} \). This fraction can be simplified by finding the greatest common divisor (GCD) of the numerator and the denominator.

• The GCD of 4 and 6 is 2.
• Divide both the numerator and the denominator by 2: \( \frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3} \).

Tackling fraction simplification with confidence requires knowing basic division and recognizing common divisors. Simplification doesn't change the value of the fraction but provides a cleaner, more understandable answer.

Properties of Limits

Understanding the properties of limits is vital for handling more complex calculus problems. These properties simplify the calculation of limits, providing easier ways to approach functions as variables move toward particular points. Here are some of the fundamental properties of limits that help us:

• Sum rule: The limit of a sum is the sum of the limits. If \( \lim_{x \to a} f(x) \) and \( \lim_{x \to a} g(x) \) exist, then \( \lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \).
• Product rule: The limit of a product is the product of the limits. If \( \lim_{x \to a} f(x) \) and \( \lim_{x \to a} g(x) \) exist, then \( \lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \).
• Quotient rule: The limit of a quotient is the quotient of the limits, provided the limit of the denominator isn't zero. If \( \lim_{x \to a} f(x) \) exists and \( \lim_{x \to a} g(x) eq 0 \), then \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \).

In our exercise, these properties allowed us to compute the limit by directly substituting and then simplifying, culminating in a straightforward and simple solution.