Problem 44 - Calculus for Biology and Medicine

Question

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow+2}\left(\frac{x^{2}}{2}-\frac{2}{x^{2}}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Apply the Limit Laws Separately to Each Term

To evaluate \( \lim_{x \to +2} \left( \frac{x^2}{2} - \frac{2}{x^2} \right) \), apply the limit laws to each term individually. The expression can be broken down as follows: \( \lim_{x \to +2} \frac{x^2}{2} \) and \( \lim_{x \to +2} \frac{2}{x^2} \).

Step 2: Evaluate the First Limit

For the first term, \( \lim_{x \to +2} \frac{x^2}{2} \), plug \( x = 2 \) directly into the expression: \( \frac{(2)^2}{2} = \frac{4}{2} = 2 \).

Step 3: Evaluate the Second Limit

For the second term, \( \lim_{x \to +2} \frac{2}{x^2} \), substitute \( x = 2 \) into the expression: \( \frac{2}{(2)^2} = \frac{2}{4} = \frac{1}{2} \).

Step 4: Combine the Results

Combine the results from Steps 2 and 3 to find the overall limit. It becomes \( 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \).

Key Concepts

limit laws

Understanding limit laws is crucial when evaluating limits in calculus. These laws provide a structured way to break down complex limit problems into simpler parts. By applying these rules, we can evaluate each term separately and recombine them to find the overall limit. Here are a few key limit laws that are often used:

• Sum and Difference Law: The limit of a sum or difference is the sum or difference of the limits. For instance, \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \).
• Product Law: If you have a product of two functions, the limit of the product is the product of the limits. It looks like \( \lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \).
• Quotient Law: The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero. This is represented as \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \) if \( \lim_{x \to a} g(x) eq 0 \).

Using these laws wisely can greatly simplify the problem solving in calculus by making it easier to handle each component of a function individually. They help us reach correct solutions systematically.

limits in calculus

Limits are foundational in calculus and describe the behavior of a function as the input approaches a certain value. When performing limit evaluation, we're trying to figure out what the output of a function gets closer to as the input nears a specified point. It's like predicting where the function is heading even if it never actually arrives there.

In the exercise, we were tasked to find \( \lim_{x \to +2}\left(\frac{x^{2}}{2}-\frac{2}{x^{2}}\right) \). The key idea was to understand how each part of the expression behaves as \( x \) approaches 2. Remember:

• Direct Substitution Method: When the function is continuous at the point we are approaching (like in many polynomial and rational functions), we can just plug the value in directly.
• Continuous Functions: Functions like \( x^2 \) are continuous, which means applying the limit to such functions is straightforward.

Grasping these concepts aids in evaluating limits effectively and lays the groundwork for understanding more advanced calculus topics, such as derivatives and integrals.

problem solving in calculus

Problem solving in calculus often involves breaking down complex expressions into manageable parts. Evaluating the given limit required using problem solving strategies that align with limit laws. Here's how to effectively tackle such problems:

• Break Down the Problem: Separate the components of the expression. In our case, \( \lim_{x \to +2} \left( \frac{x^2}{2} - \frac{2}{x^2} \right) \) was broken into two parts: \( \lim_{x \to +2} \frac{x^2}{2} \) and \( \lim_{x \to +2} \frac{2}{x^2} \).
• Use Limit Laws: Apply each relevant limit law to simplify each component. Evaluate them independently before combining them again.
• Combine Results and Simplify: After finding the limits of the individual parts, combine the results and simplify to get the final answer, which in this case was \( \frac{3}{2} \).

These strategies make calculus problems less intimidating and help develop a structured approach to problem solving. Moreover, understanding these steps improves your ability to tackle more advanced problems in the future.