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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?

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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.

Evaluate the limit \( \lim_{{h \to 0}} \frac{{\sin(x+h) - \sin(x)}}{h} \)

Understand how to evaluate the limit \( \lim_{{h \to 0}} \frac{{\sin(x+h) - \sin(x)}}{h} \) to find the derivative of the sine function. Learn key concepts in calculus, limits, and trigonometry for Grades 11-12.

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