Problem 2 - Calculus for Biology and Medicine

Question

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4} $$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{4} \).

Step 1: Identify the Indeterminate Form

First, evaluate the limit as \( x \rightarrow 2 \) by directly substituting \( x = 2 \) into the expression \( \frac{x-2}{x^2-4} \). Both the numerator \( x-2 \) and the denominator \( x^2-4 \) become zero, resulting in a \( \frac{0}{0} \) indeterminate form. Thus, l'Hospital's Rule is applicable.

Step 2: Apply l'Hospital's Rule

Since the limit results in a \( \frac{0}{0} \) form, differentiate the numerator and the denominator separately. The derivative of the numerator \( x-2 \) is 1, and the derivative of the denominator \( x^2-4 \) is \( 2x \). Apply l'Hospital's Rule:\[\lim_{x \rightarrow 2} \frac{x-2}{x^2-4} = \lim_{x \rightarrow 2} \frac{1}{2x}\]

Step 3: Evaluate the New Limit

Substitute \( x = 2 \) into the differentiated expression \( \frac{1}{2x} \):\[\lim_{x \rightarrow 2} \frac{1}{2x} = \frac{1}{2 \times 2} = \frac{1}{4}\]Thus, the limit is \( \frac{1}{4} \).

Key Concepts

Limits in Calculus

Limits are fundamental to calculus because they help us understand the behavior of functions as they approach a specific point or even infinity. When evaluating limits, we observe what happens to a function as its input value gets closer and closer to a certain number, but doesn't actually reach that number. For example, consider evaluating the limit \( \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4} \). Direct substitution of \( x = 2 \) results in the form \( \frac{0}{0} \), which doesn't provide useful information about the function's behavior. This indicates a need for further exploration or a different method.There are several techniques to find limits:

• Direct Substitution: Plugging numbers directly into the function to find an answer. If this leads to an indeterminate form, other methods are needed.
• Simplification: Sometimes, expressions can be algebraically simplified, making it easier to determine limits.
• Special Functions: Recognizing patterns like those in trigonometric functions or roots can also lead to simpler limit crossings.

Understanding these concepts is essential since they form the basis for more advanced topics like derivatives and integrals.

Indeterminate Forms

Indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) occur in calculus when evaluating the limit of a function results in expressions that are not immediately solvable. These forms hint at a deeper insight that isn't visible at first glance. Indeterminate forms are crucial because they point out cases where the straightforward evaluation isn't conclusive or helpful.Common indeterminate forms include but are not limited to:

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)
• \(0 \cdot \infty \)
• \( \infty - \infty \)
• \( 0^{0} \)
• \( 1^{\infty} \)

L'Hospital's Rule is a key tool for resolving indeterminate forms, allowing us to differentiate the numerator and denominator separately until a determinate form is obtained. For instance, in the exercise provided, applying l'Hospital's Rule turns the original indeterminate form \( \frac{0}{0} \) into a much simpler form, \( \frac{1}{2x} \), which is easily solvable.

Differentiation in Calculus

Differentiation in calculus refers to finding the derivative of a function, which reveals the rate at which the function's value changes as its variable changes. It is a cornerstone of calculus, often represented using prime notation \( f'(x) \) or Leibniz notation \( \frac{dy}{dx} \).When dealing with limits, derivatives are particularly useful. Differentiation helps us solve problems where direct substitution in limits leads to indeterminate forms. In these situations, applying l'Hospital's Rule requires differentiating the numerator and the denominator.For example, given the expression \( \frac{x-2}{x^2-4} \), direct substitution gave us \( \frac{0}{0} \). Here, taking the derivatives:

• Numerator: The derivative of \( x-2 \) is 1, as it's a linear function.
• Denominator: The derivative of \( x^2-4 \) is \( 2x \), using the power rule \( \frac{d}{dx}[x^n] = nx^{n-1} \).

Differentiation transforms the problem into one that is easily manageable, allowing for a straightforward evaluation of limits. This makes differentiation a powerful technique not just for calculus, but for solving a wide range of mathematical problems.