Evaluating Limits Using Limit Laws A Step-by-Step Solution

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In calculus, limits form the foundational concept upon which derivatives and integrals are built. Understanding how to evaluate limits is crucial for mastering calculus. The limit of a function describes the value that the function approaches as the input approaches a certain value. Limit laws provide a set of rules that simplify the process of evaluating limits. In this article, we will explore these limit laws through a detailed example, demonstrating how they can be applied to find the limit of a complex function. Our focus will be on the scenario where we are given the limits of two functions, f(x) and g(x), as x approaches a specific value, and we aim to find the limit of a combination of these functions. This method is extensively used in calculus to break down complex limits into simpler, manageable parts. This guide aims to provide a clear and comprehensive explanation of how to use these laws, ensuring a solid understanding for anyone studying calculus. We will delve into the step-by-step process of applying these laws, making it easier to tackle similar problems. Understanding these principles is not just about solving problems; it's about grasping the fundamental nature of calculus and how functions behave as they approach certain points.

Limit Laws

Before diving into the problem, let's briefly discuss the limit laws that will be used. These laws allow us to break down complex limits into simpler ones. The key limit laws 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 essential tools in evaluating limits, as they allow us to simplify complex expressions into more manageable components. By understanding and applying these laws, we can systematically solve a wide range of limit problems. This approach not only makes the process easier but also provides a clear and logical way to arrive at the solution. The following sections will demonstrate how these laws can be applied in practice to solve a specific limit problem.

Problem Statement

Let's consider a specific problem to illustrate the application of limit laws. Suppose we are given that $\lim_x \to 7} f(x) = 6$ and $\lim_{x \to 7} g(x) = 2$. Our goal is to find the limit $\lim_{x \to 7 \frac{f(x) + g(x)}{2g(x)}$. This problem is a classic example of how limit laws can be used to evaluate the limit of a function composed of other functions. The expression inside the limit involves a combination of f(x) and g(x), both of which have known limits as x approaches 7. By applying the appropriate limit laws, we can break down this complex limit into simpler parts and ultimately find the solution. This type of problem is fundamental in calculus and appears frequently in introductory courses. Mastering the technique to solve it will provide a strong foundation for more advanced topics. In the following sections, we will go through the step-by-step solution, highlighting the application of each limit law involved.

Step-by-Step Solution

To evaluate the limit $\lim_{x \to 7} \frac{f(x) + g(x)}{2g(x)}$, we will systematically apply the limit laws. This step-by-step approach will not only help us find the solution but also provide a clear understanding of how each law is utilized.

Step 1: Apply the Quotient Law

The Quotient Law states that the limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero. In our case, we can rewrite the limit as:

limxto7fracf(x)+g(x)2g(x)=fraclimxto7[f(x)+g(x)]limxto7[2g(x)]\\lim_{x \\to 7} \\frac{f(x) + g(x)}{2g(x)} = \\frac{\\lim_{x \\to 7} [f(x) + g(x)]}{\\lim_{x \\to 7} [2g(x)]}

This step is crucial as it separates the limit of the entire expression into the limits of the numerator and the denominator. Applying the Quotient Law allows us to deal with the numerator and denominator separately, making the problem more manageable. It's important to verify that the limit of the denominator is not zero before applying this law. In our case, since $\lim_{x \to 7} g(x) = 2$, the limit of the denominator will be 2 times 2, which is not zero, so we can proceed with this step.

Step 2: Apply the Sum Law

The Sum Law states that the limit of a sum is the sum of the limits. Applying this law to the numerator, we get:

limxto7[f(x)+g(x)]=limxto7f(x)+limxto7g(x)\\lim_{x \\to 7} [f(x) + g(x)] = \\lim_{x \\to 7} f(x) + \\lim_{x \\to 7} g(x)

This step further simplifies the problem by breaking down the limit of the sum into the sum of individual limits. We now have the limits of f(x) and g(x) separately, which are given in the problem statement. This is a common strategy in evaluating limits – to decompose complex expressions into simpler terms whose limits are known or easier to compute. By applying the Sum Law, we are able to isolate the limits of f(x) and g(x), which is a significant step towards finding the final answer.

Step 3: Apply the Constant Multiple Law

The Constant Multiple Law states that the limit of a constant times a function is the constant times the limit of the function. Applying this law to the denominator, we have:

limxto7[2g(x)]=2limxto7g(x)\\lim_{x \\to 7} [2g(x)] = 2 \\lim_{x \\to 7} g(x)

This step is essential for simplifying the denominator. By extracting the constant 2, we are left with the limit of g(x), which is given. The Constant Multiple Law is a powerful tool for dealing with constants within limits, as it allows us to move them outside the limit operation. This simplifies the expression and makes it easier to substitute the known limit values. Applying this law in this context helps us isolate the limit of g(x), making the subsequent calculations straightforward.

Step 4: Substitute the Given Limits

Now that we have simplified the expression using the limit laws, we can substitute the given limits: $\lim_{x \to 7} f(x) = 6$ and $\lim_{x \to 7} g(x) = 2$. Substituting these values, we get:

fraclimxto7f(x)+limxto7g(x)2limxto7g(x)=frac6+22cdot2\\frac{\\lim_{x \\to 7} f(x) + \\lim_{x \\to 7} g(x)}{2 \\lim_{x \\to 7} g(x)} = \\frac{6 + 2}{2 \\cdot 2}

This step is the culmination of the previous simplifications. By substituting the known limits, we transform the limit expression into a simple arithmetic expression. This is a crucial step in solving limit problems, as it converts the abstract concept of a limit into a concrete numerical value. The substitution is only valid after we have applied the limit laws to break down the original limit into its constituent parts. In this case, the substitution leads us to a straightforward calculation, which will give us the final answer.

Step 5: Simplify the Expression

Finally, we simplify the arithmetic expression:

frac6+22cdot2=frac84=2\\frac{6 + 2}{2 \\cdot 2} = \\frac{8}{4} = 2

Therefore, $\lim_{x \to 7} \frac{f(x) + g(x)}{2g(x)} = 2$. This final step involves basic arithmetic operations to arrive at the numerical value of the limit. After substituting the known limits, the expression simplifies to a fraction that can be easily reduced. This step highlights the importance of accurate calculations to obtain the correct answer. The result, 2, represents the value that the function $\frac{f(x) + g(x)}{2g(x)}$ approaches as x approaches 7. This completes the solution to the problem, demonstrating the effective use of limit laws in evaluating limits.

Final Answer

By applying the limit laws step-by-step, we have found that:

limxto7fracf(x)+g(x)2g(x)=2\\lim_{x \\to 7} \\frac{f(x) + g(x)}{2g(x)} = 2

This result provides a clear and concise answer to the problem. The step-by-step solution demonstrates the power and utility of limit laws in evaluating complex limits. By breaking down the problem into smaller, manageable parts, we were able to apply the appropriate limit laws and arrive at the final answer. This process not only provides the solution but also reinforces the understanding of how limits behave and how they can be manipulated using these laws. The final answer, 2, is the value that the given function approaches as x approaches 7, and it is obtained through a systematic and logical application of calculus principles.

Conclusion

In conclusion, evaluating limits using limit laws is a fundamental skill in calculus. By understanding and applying these laws, we can simplify complex limit problems into manageable steps. This article has demonstrated how to use the Quotient Law, Sum Law, and Constant Multiple Law to find the limit of a function. The step-by-step solution not only provides the answer but also illustrates the process of breaking down a complex problem into simpler parts. Mastering these techniques is crucial for further studies in calculus and related fields. The ability to evaluate limits is the cornerstone of many advanced concepts, including derivatives, integrals, and continuity. Therefore, a solid understanding of limit laws and their applications is essential for success in calculus. The example presented in this article serves as a practical guide to applying these laws effectively. By practicing similar problems, students can develop the necessary skills to confidently tackle a wide range of limit problems.