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

문제 106

3장, 문제 106

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

검증된 단계별 안내

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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주요 개념

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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.

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