The given expression represents the derivative of the sine function using its definition as a limit. The expression is:

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

This is the definition of the derivative of $\sin(x)$ at a point $x$, using the limit definition of the derivative. To solve this limit, we use the trigonometric identity for the difference of sines:

$\sin(x+h) - \sin(x) = 2\cos\left(\frac{x+h+x}{2}\right)\sin\left(\frac{h}{2}\right)$

However, instead of using the identity, let's directly differentiate the sine function using basic calculus. The derivative of $\sin(x)$ is:

$\frac{d}{dx}[\sin(x)] = \cos(x)$

Thus:

$\lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h} = \cos(x)$

Therefore, the limit simplifies to $\cos(x)$.

Would you like more details or have any specific questions?

Here are 5 related questions to consider:

1. What is the limit definition of the derivative for any general function $f(x)$?
2. How can you use trigonometric identities to simplify limits involving sine and cosine?
3. What are the derivatives of other common trigonometric functions such as $\cos(x)$ and $\tan(x)$?
4. How does the derivative of a trigonometric function behave graphically?
5. What are the higher-order derivatives of $\sin(x)$ and $\cos(x)$?

Tip: Understanding the limit definition of the derivative is key to mastering calculus, especially for trigonometric functions.