Estimate the following limits using graphs or tables.lim⁡h→0ln⁡(1+h)h{\displaystyle\lim_{h\to0}}\frac{\ln\left(1+h\right)}{h}

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

문제 2.41

2장, 문제 2.41

Estimate the following limits using graphs or tables.
$\lim_{h \to 0} \frac{\ln (1 + h)}{h}$

검증된 단계별 안내

Recognize that the limit $ \lim_{h \to 0} \frac{\ln(1+h)}{h} $ is a standard limit that can be evaluated using the definition of the derivative.

Identify that the expression $ \frac{\ln(1+h)}{h} $ resembles the difference quotient for the derivative of $ \ln(x) $ at $ x = 1 $.

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

Conclude that the limit $ \lim_{h \to 0} \frac{\ln(1+h)}{h} $ is equal to the derivative of $ \ln(x) $ at $ x = 1 $, which is 1.

Alternatively, use L'Hôpital's Rule, which applies to limits of the form $ \frac{0}{0} $, by differentiating the numerator and the denominator separately and then taking the limit.

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주요 개념

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Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, we are interested in the limit of the function ln(1+h)/h as h approaches 0. Understanding limits is crucial for analyzing the behavior of functions near specific points, especially when direct substitution may lead to indeterminate forms.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a key function in calculus, particularly in relation to growth rates and areas under curves. In the limit expression given, ln(1+h) captures the behavior of logarithmic growth as h approaches 0, which is essential for evaluating the limit.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule can be applied to the limit in the question, allowing for a more straightforward evaluation of the limit as h approaches 0.

Estimate the following limits using graphs or tables.limh→0ln(1+h)h{\(\displaystyle\]\lim\)_{h\(\to\)0}}\(\frac{\ln\left(1+h\right)}{h}\)