Evaluating The Limit Lim (b→8) [ (1/b - 1/8) / (b - 8) ] A Comprehensive Guide

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Evaluate the limit lim (b→8) [ (1/b - 1/8) / (b - 8) ]

Introduction

In the realm of calculus, evaluating limits is a fundamental concept that lays the groundwork for understanding continuity, derivatives, and integrals. This article delves into a detailed explanation of how to evaluate the limit: lim (b→8) [ (1/b - 1/8) / (b - 8) ]. We will explore various approaches, including algebraic manipulation and the concept of derivatives, to arrive at the solution. Understanding this specific limit provides valuable insights into techniques applicable to a wide range of limit problems.

The limit we are tasked with evaluating is: lim (b→8) [ (1/b - 1/8) / (b - 8) ]. This expression represents the behavior of the function f(b) = (1/b - 1/8) / (b - 8) as 'b' approaches 8. Directly substituting b = 8 into the function results in an indeterminate form (0/0), indicating that further analysis is required. This indeterminate form is a classic signal that algebraic manipulation or other techniques are necessary to determine the true limit. The presence of this indeterminate form doesn't mean the limit doesn't exist; rather, it suggests that the function's behavior near b = 8 requires a closer look. Our goal is to transform the expression into a form where the limit can be easily evaluated, often by eliminating the indeterminate form. This process typically involves algebraic simplification, such as finding a common denominator and canceling out terms. The techniques we employ here are not just specific to this problem but are widely applicable in calculus for dealing with limits, derivatives, and other related concepts. By mastering these methods, you'll be well-equipped to tackle more complex problems involving limits and understand the underlying principles that govern function behavior near specific points.

Algebraic Manipulation

The most common and often the most straightforward approach to evaluating limits like this is through algebraic manipulation. Our primary goal is to simplify the expression and eliminate the indeterminate form (0/0) that arises from direct substitution. The key to simplifying this particular expression lies in finding a common denominator for the fractions in the numerator and then attempting to cancel out terms.

Let's break down the steps:

  1. Find a Common Denominator: The numerator of our expression is (1/b - 1/8). To combine these fractions, we need a common denominator, which is the least common multiple of 'b' and 8, namely 8b. Rewriting the fractions with this common denominator gives us:

    (1/b) * (8/8) - (1/8) * (b/b) = 8/(8b) - b/(8b)

  2. Combine the Fractions: Now that the fractions have a common denominator, we can combine them:

    (8 - b) / (8b)

  3. Rewrite the Original Expression: Substituting this back into our original limit expression, we have:

    lim (b→8) [ ((8 - b) / (8b)) / (b - 8) ]

  4. Simplify the Complex Fraction: To simplify the complex fraction, we can rewrite the division as multiplication by the reciprocal:

    lim (b→8) [ (8 - b) / (8b) * (1 / (b - 8)) ]

  5. Factor out -1: Notice that (8 - b) is the negative of (b - 8). We can factor out a -1 from the numerator:

    lim (b→8) [ -1(b - 8) / (8b * (b - 8)) ]

  6. Cancel Common Factors: Now we can cancel the (b - 8) terms in the numerator and denominator, as long as b ≠ 8 (which is the case since we are considering the limit as b approaches 8):

    lim (b→8) [ -1 / (8b) ]

  7. Evaluate the Limit: With the expression simplified, we can now directly substitute b = 8:

    -1 / (8 * 8) = -1 / 64

Therefore, the limit lim (b→8) [ (1/b - 1/8) / (b - 8) ] is -1/64. This algebraic approach highlights the importance of simplifying expressions to eliminate indeterminate forms and reveal the true behavior of the function near the limit point.

Connection to Derivatives

Interestingly, the limit we just evaluated has a deep connection to the concept of derivatives in calculus. Recognizing this connection provides an alternative method for solving the problem and strengthens our understanding of the fundamental relationship between limits and derivatives. The limit in question closely resembles the definition of the derivative.

Recall the definition of the derivative of a function f(x) at a point x = a:

f'(a) = lim (h→0) [ (f(a + h) - f(a)) / h ]

Our limit can be rewritten to fit this form. Let's consider the function f(b) = 1/b. We want to find the derivative of this function at b = 8. The limit we are evaluating is:

lim (b→8) [ (1/b - 1/8) / (b - 8) ]

This can be seen as the derivative of f(b) = 1/b at b = 8. To make this more explicit, let's rewrite the limit using a substitution. Let h = b - 8. Then, b = 8 + h. As b approaches 8, h approaches 0. Substituting these into our limit, we get:

lim (h→0) [ (1/(8 + h) - 1/8) / h ]

This is precisely the definition of the derivative of f(b) = 1/b evaluated at b = 8. Now, we can find the derivative of f(b) = 1/b using the power rule. Rewriting f(b) as b^(-1), we have:

f'(b) = -1 * b^(-2) = -1/b^2

Evaluating this derivative at b = 8 gives us:

f'(8) = -1 / (8^2) = -1 / 64

This confirms our result from the algebraic manipulation. The limit lim (b→8) [ (1/b - 1/8) / (b - 8) ] is indeed the derivative of f(b) = 1/b evaluated at b = 8, and its value is -1/64. This connection between limits and derivatives is a cornerstone of calculus. Understanding this relationship not only provides alternative problem-solving techniques but also deepens our comprehension of the fundamental principles that govern rates of change and function behavior.

L'Hôpital's Rule

Another powerful tool for evaluating limits, especially those that result in indeterminate forms, is L'Hôpital's Rule. This rule is particularly useful when dealing with limits of the form 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) as x approaches a results in an indeterminate form, and if f(x) and g(x) are differentiable, then:

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

provided the limit on the right-hand side exists.

Let's apply L'Hôpital's Rule to our limit: lim (b→8) [ (1/b - 1/8) / (b - 8) ].

  1. Identify f(b) and g(b): In our case, f(b) = (1/b - 1/8) and g(b) = (b - 8).

  2. Check for Indeterminate Form: As we saw earlier, direct substitution of b = 8 into the expression results in the indeterminate form 0/0.

  3. Find the Derivatives: We need to find the derivatives of f(b) and g(b) with respect to b:

    • f'(b) = d/db (1/b - 1/8) = -1/b^2 (since the derivative of a constant, 1/8, is 0)
    • g'(b) = d/db (b - 8) = 1

  4. Apply L'Hôpital's Rule: Now we can apply L'Hôpital's Rule:

    lim (b→8) [ (1/b - 1/8) / (b - 8) ] = lim (b→8) [ (-1/b^2) / 1 ]

  5. Evaluate the Limit: Substitute b = 8 into the new expression:

    lim (b→8) [ -1/b^2 ] = -1 / (8^2) = -1 / 64

Again, we arrive at the same result, -1/64. L'Hôpital's Rule provides a direct and often efficient method for evaluating limits involving indeterminate forms. However, it's crucial to ensure that the conditions for applying the rule are met, namely, that the limit results in an indeterminate form and that the functions are differentiable. This rule is a valuable addition to our toolkit for evaluating limits and provides a different perspective on the problem.

Conclusion

In this article, we have thoroughly evaluated the limit lim (b→8) [ (1/b - 1/8) / (b - 8) ] using three different methods: algebraic manipulation, connection to derivatives, and L'Hôpital's Rule. Each approach provides a unique perspective on the problem and reinforces the fundamental concepts of calculus. Algebraic manipulation allowed us to simplify the expression by finding a common denominator and canceling out terms, ultimately leading to the result -1/64. Recognizing the limit as the derivative of f(b) = 1/b at b = 8 offered an alternative solution path and highlighted the relationship between limits and derivatives. Finally, L'Hôpital's Rule provided a direct method for evaluating the limit by taking the derivatives of the numerator and denominator, again confirming the result of -1/64.

This exploration underscores the importance of mastering various techniques for evaluating limits. The ability to approach a problem from different angles not only enhances problem-solving skills but also deepens the understanding of the underlying mathematical principles. Limits are the building blocks of calculus, and a solid grasp of limit evaluation techniques is crucial for success in more advanced topics such as continuity, differentiation, and integration. By understanding these methods, you are well-equipped to tackle a wide range of calculus problems and appreciate the elegance and power of mathematical analysis. This specific example serves as a valuable case study for applying these techniques and solidifying your understanding of limit evaluation.