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.