To evaluate the limit

$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h},$

we recognize that this limit is the definition of the derivative of the function $f(x)$ at the point $x$. Thus, the limit evaluates to $f'(x)$, which is the derivative of $f(x)$ with respect to $x$.

The specific value of the limit depends on the form of the function $f(x)$. If you provide the function $f(x)$, I can calculate the exact value of the derivative at the point $x$.

Would you like to provide the function $f(x)$ or go over an example?

Here are some related questions:

1. How do we find the derivative of $f(x) = x^2$ using this limit definition?
2. What is the derivative of trigonometric functions like $f(x) = \sin(x)$?
3. How does the derivative change if the function $f(x)$ is a polynomial?
4. What is the physical interpretation of the derivative in terms of velocity and acceleration?
5. How can you check if the limit definition correctly gives the derivative?

Tip: When using the limit definition to find a derivative, expanding $f(x+h)$ using series expansion or known identities can simplify the process significantly.