L'Hopital's Rule Evaluate Limit Of (e^x + Cos(x) - X - 2) / (x^3 - 4x^2)

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In this comprehensive guide, we will evaluate the limit of the function (e^x + cos(x) - x - 2) / (x^3 - 4x^2) as x approaches 0. We will employ L'Hopital's Rule, a powerful tool for handling indeterminate forms, and meticulously demonstrate each step with proper notation. Understanding limits is a cornerstone of calculus, and mastering techniques like L'Hopital's Rule is crucial for solving complex problems in mathematics, physics, and engineering. This article aims to provide a clear and detailed explanation, ensuring readers can confidently apply these concepts to similar problems. To fully grasp the intricacies of this problem, it's essential to have a solid foundation in differential calculus, including the derivatives of exponential and trigonometric functions. We will begin by directly substituting x = 0 into the function and observing the result. This initial step is vital as it reveals whether the limit can be evaluated directly or if an indeterminate form arises, necessitating the use of L'Hopital's Rule. The ability to identify indeterminate forms is a key skill in limit evaluation, and this article will highlight the significance of this step. Furthermore, we will emphasize the importance of verifying the conditions under which L'Hopital's Rule can be applied. Misapplication of this rule can lead to incorrect results, so a thorough understanding of its prerequisites is essential. By following this step-by-step approach, readers will not only learn how to solve this specific problem but also gain a deeper understanding of the underlying principles and techniques involved in limit evaluation. The ultimate goal is to empower readers with the knowledge and skills necessary to tackle a wide range of limit problems with confidence and accuracy.

1. Initial Evaluation and Indeterminate Form

To begin evaluating the limit, we first substitute x = 0 into the function:

(e^0 + cos(0) - 0 - 2) / (0^3 - 4(0)^2) = (1 + 1 - 0 - 2) / (0 - 0) = 0 / 0

This result, 0/0, is an indeterminate form. This means we cannot directly determine the limit by substitution and need to employ a different technique. The appearance of this indeterminate form signals that L'Hopital's Rule may be applicable. L'Hopital's Rule is specifically designed to handle indeterminate forms such as 0/0 and ∞/∞. However, it's crucial to remember that L'Hopital's Rule is not a universal solution for all limit problems. It only applies under specific conditions, which we will discuss in detail later. The initial substitution step is a critical part of the process, as it dictates the subsequent steps. Had we obtained a determinate form, such as a finite number or infinity, the evaluation would have been complete at this stage. Recognizing the indeterminate form is a fundamental skill in calculus, and it's essential to develop this ability to solve limit problems effectively. Understanding the concept of indeterminate forms is not just about applying a rule; it's about understanding the behavior of functions as they approach certain values. This understanding forms the basis for more advanced concepts in calculus, such as continuity and differentiability. In the next section, we will delve into the conditions required for applying L'Hopital's Rule and verify that they are met in this case. This will pave the way for the actual application of the rule and the subsequent evaluation of the limit.

2. Applying L'Hopital's Rule: First Application

Since we have the indeterminate form 0/0, we can apply L'Hopital's Rule. L'Hopital's Rule states that if the limit of f(x)/g(x) as x approaches a results in an indeterminate form 0/0 or ∞/∞, and if the limit of f'(x)/g'(x) exists, then:

lim (x→a) f(x)/g(x) = lim (x→a) f'(x)/g'(x)

In our case, f(x) = e^x + cos(x) - x - 2 and g(x) = x^3 - 4x^2. We need to find the derivatives of both functions:

f'(x) = d/dx (e^x + cos(x) - x - 2) = e^x - sin(x) - 1 g'(x) = d/dx (x^3 - 4x^2) = 3x^2 - 8x

Now, we apply L'Hopital's Rule and find the limit of the ratio of the derivatives:

lim (x→0) (e^x - sin(x) - 1) / (3x^2 - 8x)

We again substitute x = 0 into the new expression:

(e^0 - sin(0) - 1) / (3(0)^2 - 8(0)) = (1 - 0 - 1) / (0 - 0) = 0 / 0

We still have the indeterminate form 0/0. This means we need to apply L'Hopital's Rule again. The repeated application of L'Hopital's Rule is a common technique when dealing with certain types of indeterminate forms. It's crucial to remember that each application of the rule requires verifying that the conditions for its application are still met. In this case, we again have the 0/0 form, which justifies another application. The process of finding the derivatives is a fundamental aspect of applying L'Hopital's Rule. A strong understanding of differentiation rules is essential for accurate calculations. Common mistakes in applying L'Hopital's Rule often stem from incorrect differentiation. Therefore, it's vital to practice and master differentiation techniques. In the next step, we will apply L'Hopital's Rule for the second time, finding the derivatives of the numerator and denominator again and evaluating the limit of the resulting expression. This iterative process will continue until we obtain a determinate form or determine that the limit does not exist.

3. Applying L'Hopital's Rule: Second Application

Since we still have the indeterminate form 0/0, we apply L'Hopital's Rule again. We need to find the derivatives of f'(x) and g'(x):

f''(x) = d/dx (e^x - sin(x) - 1) = e^x - cos(x) g''(x) = d/dx (3x^2 - 8x) = 6x - 8

Now, we apply L'Hopital's Rule again:

lim (x→0) (e^x - cos(x)) / (6x - 8)

We substitute x = 0 into the new expression:

(e^0 - cos(0)) / (6(0) - 8) = (1 - 1) / (0 - 8) = 0 / -8 = 0

This time, we obtain a determinate form, 0/-8, which simplifies to 0. Therefore, the limit exists and is equal to 0. The successful application of L'Hopital's Rule twice highlights the power of this technique in resolving indeterminate forms. It's important to note that the process of repeatedly applying L'Hopital's Rule can sometimes be necessary to arrive at a determinate form. However, it's also crucial to be mindful of the complexity of the derivatives involved. If the derivatives become increasingly complex with each application, it might be worth considering alternative methods for evaluating the limit. The fact that we obtained a determinate form after two applications indicates that the original function has a well-defined limit as x approaches 0. This limit represents the value that the function approaches as x gets arbitrarily close to 0. Understanding the concept of a limit is fundamental to calculus and has wide-ranging applications in various fields. In this case, the limit being 0 suggests that the function's value becomes negligibly small as x approaches 0. This information can be valuable in analyzing the behavior of the function and its relationship to other functions. In the next section, we will summarize the steps taken and state the final answer.

4. Final Answer and Conclusion

After evaluating the limit using L'Hopital's Rule twice, we have arrived at the final answer. The steps we took were:

  1. Initial substitution of x = 0, which resulted in the indeterminate form 0/0.
  2. Application of L'Hopital's Rule, finding the derivatives of the numerator and denominator.
  3. Substitution of x = 0 again, which still resulted in the indeterminate form 0/0.
  4. Second application of L'Hopital's Rule, finding the derivatives of the new numerator and denominator.
  5. Final substitution of x = 0, which resulted in the determinate form 0/-8.

Therefore, the limit is:

lim (x→0) (e^x + cos(x) - x - 2) / (x^3 - 4x^2) = 0

In conclusion, we successfully evaluated the limit using L'Hopital's Rule. This problem demonstrates the importance of recognizing indeterminate forms and applying appropriate techniques to resolve them. L'Hopital's Rule is a powerful tool, but it's crucial to understand its conditions and apply it correctly. The ability to evaluate limits is a fundamental skill in calculus and has wide-ranging applications in mathematics, physics, and engineering. This example highlights the step-by-step process involved in applying L'Hopital's Rule, emphasizing the importance of careful differentiation and substitution. Understanding the concept of limits is essential for grasping more advanced calculus concepts, such as continuity, differentiability, and integration. The process of evaluating limits often involves a combination of algebraic manipulation, differentiation techniques, and the application of specific rules like L'Hopital's Rule. By mastering these techniques, students can confidently tackle a wide range of limit problems and gain a deeper understanding of the fundamental principles of calculus. This article has provided a detailed and comprehensive guide to solving this particular limit problem, and the principles and techniques discussed can be applied to similar problems in the future.