Evaluating Limits A Step-by-Step Guide To Lim X→5 [f(x) + G(x)] / [2g(x)]

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In the realm of calculus, limits form the bedrock upon which many advanced concepts are built. Understanding how to evaluate limits is crucial for grasping continuity, derivatives, and integrals. In this comprehensive guide, we will delve into the evaluation of a specific limit problem, providing a step-by-step solution and elucidating the underlying principles. This article aims to provide an in-depth understanding of limit evaluation, catering to both students and enthusiasts seeking to master this fundamental concept.

Let's consider the problem at hand: Given that lim x→5 f(x) = 7 and lim x→5 g(x) = 9, our objective is to find the limit of the expression [f(x) + g(x)] / [2g(x)] as x approaches 5. This problem exemplifies the application of limit laws, which are essential tools for simplifying and evaluating complex limits. The key to solving this problem lies in understanding and applying these limit laws systematically. We will break down the problem into manageable parts, making it easier to grasp each step involved in the solution.

Limit Laws: The Foundation of Evaluation

Before diving into the solution, it's crucial to understand the fundamental limit laws that govern the behavior of limits. These laws allow us to manipulate limits of complex expressions by breaking them down into simpler components. The relevant laws for this problem include:

  1. Sum/Difference Law: The limit of a sum (or difference) is the sum (or difference) of the limits.
  2. Constant Multiple Law: The limit of a constant times a function is the constant times the limit of the function.
  3. Quotient Law: The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero.

These laws are the building blocks for evaluating more complex limits, and mastering them is essential for success in calculus. In the following sections, we will apply these laws to solve the given problem systematically.

Step-by-Step Solution: Applying Limit Laws

Now, let's embark on the journey of solving the limit problem step by step. Our goal is to find:

lim x→5 [f(x) + g(x)] / [2g(x)]

Step 1: Applying the Quotient Law

The first step is to apply the quotient law, which states that the limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero. Mathematically, this can be expressed as:

lim x→c [f(x) / g(x)] = [lim x→c f(x)] / [lim x→c g(x)], if lim x→c g(x) ≠ 0

Applying this to our problem, we get:

lim x→5 [f(x) + g(x)] / [2g(x)] = [lim x→5 (f(x) + g(x))] / [lim x→5 2g(x)]

Step 2: Applying the Sum Law

Next, we apply the sum law to the numerator. The sum law states that the limit of a sum is the sum of the limits:

lim x→c [f(x) + g(x)] = lim x→c f(x) + lim x→c g(x)

Applying this to our numerator, we have:

lim x→5 [f(x) + g(x)] = lim x→5 f(x) + lim x→5 g(x)

Step 3: Applying the Constant Multiple Law

Now, let's address the denominator. We have lim x→5 2g(x). The constant multiple law states that the limit of a constant times a function is the constant times the limit of the function:

lim x→c [k * f(x)] = k * lim x→c f(x)

Applying this law, we get:

lim x→5 2g(x) = 2 * lim x→5 g(x)

Step 4: Substituting Given Limits

We are given that lim x→5 f(x) = 7 and lim x→5 g(x) = 9. Now, we substitute these values into our expression:

[lim x→5 f(x) + lim x→5 g(x)] / [2 * lim x→5 g(x)] = [7 + 9] / [2 * 9]

Step 5: Simplifying the Expression

Finally, we simplify the expression:

[7 + 9] / [2 * 9] = 16 / 18 = 8 / 9

Therefore, the limit of the given expression as x approaches 5 is 8/9. This step-by-step solution demonstrates the power of limit laws in simplifying and evaluating complex limits.

Common Mistakes to Avoid

When working with limits, several common mistakes can lead to incorrect answers. Awareness of these pitfalls is crucial for accurate problem-solving. Here are some mistakes to avoid:

  1. Dividing by Zero: One of the most common errors is attempting to divide by zero. The quotient law is only applicable if the limit of the denominator is not zero. Always check this condition before applying the law.
  2. Incorrectly Applying Limit Laws: Applying limit laws in the wrong order or misinterpreting them can lead to incorrect results. Ensure you understand the conditions under which each law is applicable.
  3. Assuming Limits Always Exist: Not all functions have limits at every point. It's essential to verify that the limits exist before applying any limit laws.
  4. Algebraic Errors: Mistakes in algebraic manipulations can easily lead to incorrect answers. Double-check your work, especially when simplifying expressions.
  5. Ignoring Indeterminate Forms: Indeterminate forms such as 0/0 or ∞/∞ require special techniques like L'Hôpital's Rule. Simply substituting values may not yield the correct limit.

By being mindful of these common mistakes, you can significantly improve your accuracy in evaluating limits. Practice and careful attention to detail are key to mastering this skill.

Practice Problems

To solidify your understanding of limit evaluation, working through practice problems is essential. Here are a few problems similar to the one we just solved:

  1. Let lim x→2 f(x) = 4 and lim x→2 g(x) = -2. Find lim x→2 [3f(x) - g(x)] / [f(x) + 2g(x)].
  2. Given lim x→-1 f(x) = 5 and lim x→-1 g(x) = 3, evaluate lim x→-1 [f(x) * g(x)] / [g(x) - f(x)].
  3. If lim x→0 f(x) = 0 and lim x→0 g(x) = 1, find lim x→0 [2f(x) + g(x)] / [g(x)^2 + 1].

Working through these problems will help you reinforce your understanding of limit laws and their application. Remember to break down each problem into steps and carefully apply the relevant laws.

Real-World Applications of Limits

While limits may seem like an abstract concept, they have numerous real-world applications in various fields. Here are a few examples:

  1. Physics: In physics, limits are used to define instantaneous velocity and acceleration. The derivative, which is a limit, is crucial for understanding motion and dynamics.
  2. Engineering: Engineers use limits to analyze the behavior of structures under stress and to design stable and efficient systems. Limits are also used in control systems and signal processing.
  3. Economics: Limits are used in economics to model marginal cost and marginal revenue, which are essential concepts in optimizing production and pricing strategies.
  4. Computer Science: In computer science, limits are used in the analysis of algorithms and data structures. They help determine the efficiency and scalability of algorithms.
  5. Finance: Limits are used in finance to calculate compound interest and to model the behavior of financial markets. They are also used in risk management and portfolio optimization.

These examples highlight the broad applicability of limits in solving real-world problems. Understanding limits provides a powerful tool for analyzing and modeling complex systems.

In conclusion, evaluating limits is a fundamental skill in calculus with far-reaching applications. By understanding and applying limit laws systematically, complex limit problems can be simplified and solved effectively. In this guide, we have demonstrated a step-by-step solution to the problem of finding lim x→5 [f(x) + g(x)] / [2g(x)], given lim x→5 f(x) = 7 and lim x→5 g(x) = 9. We also discussed common mistakes to avoid and provided practice problems to reinforce your understanding. Remember, mastering limits is not just about memorizing laws; it's about developing a conceptual understanding and applying it to various problems. As you continue your journey in calculus, the principles you've learned here will serve as a solid foundation for more advanced topics.