limℎ→ 01+ℎ‾‾‾‾‾√−1ℎ
represents 𝑓′(𝑎) for some function 𝑓 and some some number 𝑎. State such an 𝑓 and 𝑎. Enter your answer in the form "𝑓(𝑥),𝑎", i.e. list the function 𝑓 and the value 𝑎 separated by a comma.

The given limit represents the definition of the derivative of some function \( f \) at some point \( a \):

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

Given:

\[
\lim_{h \to 0} \frac{\sqrt{1+h} - 1}{h}
\]

we need to identify \( f(x) \) and \( a \) such that this limit expression fits the definition of the derivative \( f'(a) \).

To match this with the definition of the derivative, observe the expression \(\sqrt{1+h}\) and compare it to \(f(a+h)\). 

Let’s set \( f(x) = \sqrt{x} \).

Now, we need to find the value of \( a \) such that \( f(a) = 1 \).

Since \( f(x) = \sqrt{x} \), we solve for \( a \) in:

\[
\sqrt{a} = 1 \implies a = 1
\]

So, \( f(x) = \sqrt{x} \) and \( a = 1 \).

Therefore, the function \( f \) and the value \( a \) are:

\[
f(x), a = \sqrt{x}, 1
\]

So the answer is:

\[
\boxed{\sqrt{x}, 1}
\]

Would you like more details or have any questions? Here are five related questions to explore further:

1. How do you compute the derivative of \( f(x) = \sqrt{x} \) using the limit definition?
2. What is the geometric interpretation of the derivative at a point \( a \)?
3. How does the derivative of a function at a point relate to the tangent line at that point?
4. Can you generalize the derivative of \( f(x) = \sqrt{x} \) for any \( x \)?
5. What is the relationship between the limit definition of a derivative and the power rule?

**Tip:** When dealing with limits and derivatives, recognizing the form of the function and its transformation is key to simplifying and solving the expression.

limℎ→ 01+ℎ‾‾‾‾‾√−1ℎ represents 𝑓′(𝑎) for some function 𝑓 and some number 𝑎. State such an 𝑓 and 𝑎. Enter your answer in the form '𝑓(𝑥),𝑎', i.e., list the function 𝑓 and the value 𝑎 separated by a comma.

Explore how to use the limit definition of the derivative to find the function f(x) and the value a for the expression lim_{h→0} (√(1+h) - 1) / h. This problem covers calculus concepts such as limits and derivatives, making it suitable for advanced high school or early college students.

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