https://file.aiquickdraw.com/mathbot/file/1727301232250866ui420.jpg

The problem you shared is about finding a function \( f(x) \) and a value \( a \) based on the given limit:

\[
\lim_{h \to 0} \frac{\ln(e + h) - 1}{h}
\]

This limit looks like the definition of the derivative for some function evaluated at a particular point \( a \), i.e.,

\[
\lim_{h \to 0} \frac{f(a + h) - f(a)}{h} = f'(a)
\]

From the expression, we can recognize the form:

\[
\lim_{h \to 0} \frac{\ln(e + h) - 1}{h}
\]

We can see that:

- The term \( \ln(e + h) \) suggests that the function \( f(x) = \ln(x) \), since its derivative is \( \frac{1}{x} \).
- The constant 1 in the limit corresponds to \( \ln(e) \), which is the value of the function at \( x = e \). This means \( a = e \).

Thus:

- The function \( f(x) = \ln(x) \).
- The value \( a = e \).

Would you like further details on how this is derived or have any questions? Here are some related questions for practice:

1. What is the derivative of \( f(x) = \ln(x) \)?
2. How would you set up the limit definition of the derivative for another function, like \( f(x) = \sqrt{x} \)?
3. What is the importance of recognizing the form of limits as derivative definitions?
4. How does the natural logarithm function behave around \( x = e \)?
5. Could you apply the same limit process to find the derivative of \( f(x) = \sin(x) \) at some value \( a \)?

**Tip:** Always try to recognize the structure of limits that represent derivatives—it’s a useful way to solve many calculus problems quickly!

The formula lim (h -> 0) [ln(e + h) - 1] / h represents f'(a) for some function f and some number a. Find f(x) and a.

Learn how to find the function f(x) and the number a from a given limit that represents the derivative f'(a). This problem focuses on the natural logarithm function and uses the limit definition of a derivative. Suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation