Use the definition of the derivative to evaluate the following limits.lim⁡h→0ln⁡(e8+h)−8h\lim_{h\to0}\frac{\ln\$$left(e^8+h$$\right)-8}{h}_{}

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 106

Chapter 3, Problem 106

Use the definition of the derivative to evaluate the following limits.
$\lim_{h \to 0} {\frac{\ln (e^{8} + h) - 8}{h}}_{}$

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Recognize that the limit expression $ \lim_{h \to 0} \frac{\ln(e^8 + h) - 8}{h} $ is in the form of the definition of the derivative, $ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $.

Identify the function $ f(x) = \ln(x) $ and the point $ x = e^8 $.

Rewrite the expression as $ \lim_{h \to 0} \frac{\ln((e^8) + h) - \ln(e^8)}{h} $, which matches the derivative definition for $ f(x) = \ln(x) $ at $ x = e^8 $.

Recall that the derivative of $ \ln(x) $ is $ \frac{1}{x} $.

Evaluate the derivative at $ x = e^8 $ to find $ f'(e^8) = \frac{1}{e^8} $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definition of the Derivative

The derivative of a function at a point is defined as the limit of the average rate of change of the function as the interval approaches zero. Mathematically, it is expressed as f'(a) = lim(h→0) [f(a+h) - f(a)] / h. This concept is fundamental in calculus as it provides a way to determine the instantaneous rate of change of a function.

Natural Logarithm Properties

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. Key properties include ln(ab) = ln(a) + ln(b) and ln(a/b) = ln(a) - ln(b). Understanding these properties is essential for simplifying expressions involving logarithms, especially when evaluating limits.

Limit Evaluation Techniques

Limit evaluation techniques involve methods to find the value that a function approaches as the input approaches a certain point. Common techniques include direct substitution, factoring, and using L'Hôpital's Rule for indeterminate forms. Mastery of these techniques is crucial for solving problems that involve limits, particularly in the context of derivatives.

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Textbook Question

Use the definition of the derivative to evaluate the following limits.

$\lim_{x \to 2} {\frac{5^{x} - 25}{x - 2}}_{}$

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Textbook Question

The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>

Estimate the instantaneous rate of growth in 1985.

Textbook Question

Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000.

a. Determine the average and marginal costs for x = 3000 lawn mowers.

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Textbook Question

Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000.

b. Interpret the meaning of your results in part (a).

Textbook Question

Explain why the average rate of growth from 1950 to 1960 is a good approximation to the (instantaneous) rate of growth in 1955.

Textbook Question

Product Rule for three functions Assume f, g, and h are differentiable at x.