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

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In the realm of calculus, understanding limits is fundamental. Limits form the bedrock upon which concepts like continuity, derivatives, and integrals are built. This article delves into the evaluation of a specific limit problem, providing a detailed, step-by-step explanation suitable for students and enthusiasts alike. We will explore how to utilize limit rules effectively to solve the problem at hand: evaluating lim x→5 [f(x) + g(x)] / [2g(x)], given that lim x→5 f(x) = 7 and lim x→5 g(x) = 9. This exploration will not only enhance your understanding of limits but also equip you with the skills to tackle similar problems with confidence. Understanding limits is crucial for grasping the behavior of functions as they approach specific points, making it an indispensable tool in mathematical analysis. The concepts we discuss here are not just theoretical; they have practical applications in various fields, including physics, engineering, and economics, where understanding rates of change and approaching values is essential. The limit concept allows us to analyze the behavior of functions at points where they might not be directly defined or where their behavior is complex. By mastering these techniques, you'll be better prepared to handle more advanced calculus topics and real-world applications.

Let’s clearly define the problem. We are tasked with finding the limit of a rational function as x approaches 5. The function in question is [f(x) + g(x)] / [2g(x)]. Crucially, we are given two essential pieces of information:

  • lim x→5 f(x) = 7
  • lim x→5 g(x) = 9

These limits tell us that as x gets closer and closer to 5, the function f(x) approaches the value 7, and the function g(x) approaches the value 9. This information is the cornerstone of our solution, as we will use the properties of limits to break down the complex expression into simpler parts that we can evaluate directly. Understanding the given information is paramount; it sets the stage for applying the limit rules. Without knowing the individual limits of f(x) and g(x), we would be unable to proceed. The problem is carefully constructed to allow us to leverage these known limits, showcasing the power of limit rules in simplifying complex evaluations. It's important to recognize that the existence of these individual limits is a prerequisite for applying many of the limit laws, highlighting the interconnected nature of calculus concepts. The clarity of the problem statement is vital for ensuring that we approach the solution methodically and accurately, avoiding common pitfalls that can arise from misinterpreting the given information.

To solve this problem, we will employ several key limit rules. These rules allow us to manipulate limits of combinations of functions, making complex limits tractable. The rules we will use are:

  • Sum Rule: The limit of a sum is the sum of the limits. That is, lim x→c [f(x) + g(x)] = lim x→c f(x) + lim x→c g(x).
  • Constant Multiple Rule: The limit of a constant times a function is the constant times the limit of the function. That is, lim x→c [k * f(x)] = k * lim x→c f(x), where k is a constant.
  • Quotient Rule: The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero. That is, lim x→c [f(x) / g(x)] = [lim x→c f(x)] / [lim x→c g(x)], if lim x→c g(x) ≠ 0.

These rules are the essential tools in our arsenal for evaluating the limit in question. The sum rule allows us to separate the limit of the sum f(x) + g(x) into the sum of individual limits. The constant multiple rule is crucial for handling the 2g(x) in the denominator. Finally, the quotient rule enables us to deal with the overall fraction, provided we ensure the limit of the denominator is not zero. Understanding the conditions under which these rules apply is just as important as knowing the rules themselves. For instance, the quotient rule is only valid if the limit of the denominator is non-zero. This highlights the importance of checking assumptions before blindly applying rules. These limit rules are not just abstract mathematical constructs; they are powerful tools that allow us to break down complex problems into simpler, manageable components. By mastering these rules, we can systematically approach a wide variety of limit problems and develop a deeper understanding of how functions behave near specific points.

Now, let's apply these rules to our specific problem:

lim x→5 [f(x) + g(x)] / [2g(x)]

  1. Apply the Quotient Rule:

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

    We can apply the quotient rule because we know that lim x→5 2g(x) will not be zero, as lim x→5 g(x) = 9, and 2 * 9 = 18, which is not zero.

  2. Apply the Sum Rule to the numerator:

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

    Using the given information, we substitute the limits of f(x) and g(x):

    lim x→5 f(x) + lim x→5 g(x) = 7 + 9 = 16

  3. Apply the Constant Multiple Rule to the denominator:

    lim x→5 2g(x) = 2 * lim x→5 g(x)

    Again, using the given information, we substitute the limit of g(x):

    2 * lim x→5 g(x) = 2 * 9 = 18

  4. Substitute the results back into the quotient:

    [lim x→5 (f(x) + g(x))] / [lim x→5 2g(x)] = 16 / 18

  5. Simplify the fraction:

    16 / 18 = 8 / 9

Therefore, lim x→5 [f(x) + g(x)] / [2g(x)] = 8 / 9. This step-by-step breakdown illustrates the power of applying limit rules systematically. By breaking down the complex limit into smaller, more manageable parts, we can easily evaluate each part using the given information and basic arithmetic. The application of the quotient rule first allows us to separate the numerator and denominator. Then, the sum rule simplifies the numerator, and the constant multiple rule handles the denominator. Finally, substituting the known limits and simplifying the resulting fraction gives us the final answer. Each step is justified by the limit rules, ensuring the correctness of the solution. This method highlights the importance of understanding and applying limit rules correctly to solve complex problems. The clear and methodical approach demonstrated here can be applied to a wide range of limit problems, building confidence and proficiency in calculus.

The final answer to the problem, lim x→5 [f(x) + g(x)] / [2g(x)], is 8/9. This result represents the value that the function [f(x) + g(x)] / [2g(x)] approaches as x gets arbitrarily close to 5. It's a precise numerical value, obtained by rigorously applying the limit rules and given information. The clarity and accuracy of the final answer underscore the importance of a methodical approach in solving limit problems. This single fraction encapsulates the entire process, summarizing the behavior of the function near the point x = 5. The significance of this answer extends beyond the immediate problem; it demonstrates the power of limits in understanding the behavior of functions and laying the groundwork for more advanced calculus concepts. This final result serves as a testament to the effectiveness of limit rules in simplifying complex expressions and providing clear, concise solutions. The journey from the initial problem statement to this final answer illustrates the beauty and precision of mathematical analysis.

In conclusion, we have successfully evaluated the limit lim x→5 [f(x) + g(x)] / [2g(x)] using the fundamental limit rules. By applying the quotient rule, sum rule, and constant multiple rule, we systematically broke down the complex limit into simpler parts. We then used the given information, lim x→5 f(x) = 7 and lim x→5 g(x) = 9, to evaluate these simpler limits. This process led us to the final answer of 8/9. This exercise showcases the importance of understanding and applying limit rules correctly. These rules are not just abstract concepts; they are powerful tools that enable us to analyze the behavior of functions as they approach specific points. The ability to manipulate limits and simplify complex expressions is crucial for success in calculus and related fields. This comprehensive walkthrough, from problem statement to final answer, provides a clear example of how to approach limit problems systematically and effectively. The skills and knowledge gained through this process are invaluable for tackling more advanced calculus topics and real-world applications. Ultimately, mastering limits is a key step in developing a strong foundation in mathematics and its applications, allowing us to explore and understand the world around us with greater precision and insight.