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Chapter 4: Problem 35

Evaluate \(\lim _{h \rightarrow 0} \frac{\ln (7+h)-\ln 7}{h}\).

Short Answer

Expert verified
The limit is \(\frac{1}{7}\).

Step by step solution

01

Identify the Limit Expression

The limit to be evaluated is \(\text{lim}_{h \to 0} \frac{\text{ln}(7+h) - \text{ln}(7)}{h}\). Recognize that it represents the derivative of a function.
02

Recognize the Logarithmic Function

Notice that \(\text{ln}(7+h)\) is a natural logarithm function where \(7+h\) is the argument.
03

Apply the Definition of Derivative

See that \(\text{lim}_{h \to 0} \frac{\text{ln}(7+h) - \text{ln}(7)}{h}\) corresponds to the definition of the derivative \(f'(a) = \text{lim}_{h \to 0} \frac{f(a+h) - f(a)}{h}\) for the function \(f(x) = \text{ln}(x)\) at \(x = 7\).
04

Use the Derivative of the Natural Logarithm

Recall that the derivative of the natural logarithm function \(f(x) = \text{ln}(x)\) is \(f'(x) = \frac{1}{x}\). Therefore, \(f'(7) = \frac{1}{7}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

limits and continuity
Understanding limits and continuity is crucial for tackling many problems in calculus. A limit helps us find the value that a function approaches as the input approaches a certain value. In this exercise, we are interested in the limit of the expression \(\frac{\text{ln}(7+h) - \text{ln}(7)}{h}\) as \(h\) approaches 0. When we talk about continuity, we mean that the function behaves smoothly without any jumps or breaks. The natural logarithm function \(\text{ln}(x)\) is continuous for all positive \(x,\) making it suitable for this limit calculation. Always ensure the function is continuous at the point of interest when applying limits.
natural logarithm differentiation
Differentiating the natural logarithm function is a common task in calculus. For a function \(f(x) = \text{ln}(x)\), its derivative is given by \(f'(x) = \frac{1}{x}.\) In this exercise, recognize that \(\text{ln}(7+h)\) and \(\text{ln}(7)\) involve the natural logarithm. The expression \(\text{lim}_{h \to 0} \frac{\text{ln}(7+h) - \text{ln}(7)}{h}\) fits the form of the derivative. Therefore, we can say that the derivative of \(\text{ln}(x)\) at \(x = 7\) is \(\frac{1}{7}\). This is a powerful and efficient way to determine the result using the properties of logarithms.
definition of the derivative
The definition of the derivative is fundamental in calculus. It represents the rate at which a function changes at a particular point. Mathematically, the derivative of a function \(f(x)\) at a point \(a\) is given by \(\text{lim}_{h \to 0} \frac{f(a+h) - f(a)}{h}.\) In this problem, \(f(x) = \text{ln}(x)\) and we evaluate at \(a = 7.\) So our limit expression \(\text{lim}_{h \to 0} \frac{\text{ln}(7+h) - \text{ln}(7)}{h}\) directly uses the definition of the derivative. Plugging \(7\) into the derivative of \(\text{ln}(x)\), we get \(\frac{1}{7}.\) Understanding this definition allows you to approach similar problems confidently.

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