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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 2 - LimitsProblema 2.41
Capitolo 2, Problema 2.41

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

Guida verificata passo dopo passo
1
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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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.
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One-Sided Limits

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.
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Derivative of the Natural Logarithmic Function

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.
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Power Rules
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Use an appropriate limit definition to prove the following limits.


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