To evaluate the limit

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

we can apply a standard trigonometric identity and the definition of a derivative.

Step-by-step process:

1. Apply the sine addition formula: $\sin(x + h) = \sin(x)\cos(h) + \cos(x)\sin(h).$ Substituting this into the limit expression: $\lim_{h \to 0} \frac{[\sin(x)\cos(h) + \cos(x)\sin(h)] - \sin(x)}{h}.$

2. Simplify the numerator: $= \lim_{h \to 0} \frac{\sin(x)(\cos(h) - 1) + \cos(x)\sin(h)}{h}.$

3. Split the limit into two parts: $= \lim_{h \to 0} \left[\sin(x) \cdot \frac{\cos(h) - 1}{h} + \cos(x) \cdot \frac{\sin(h)}{h}\right].$

4. Evaluate the limits using standard results:

   • $\lim_{h \to 0} \frac{\sin(h)}{h} = 1$,
   • $\lim_{h \to 0} \frac{\cos(h) - 1}{h} = 0$.

   Substituting these results, we get: $\lim_{h \to 0} \left[\sin(x) \cdot 0 + \cos(x) \cdot 1\right] = \cos(x).$

Final result:

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

This is the derivative of $\sin(x)$, confirming the limit represents $\frac{d}{dx}[\sin(x)] = \cos(x)$.

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Would you like more details on any of the steps?

Here are 5 related questions to explore further:

1. What is the limit of $(\cos(x+h)-\cos(x))/h$ as $h \to 0$?
2. How can we apply the definition of derivative to find the derivative of $\cos(x)$?
3. What is the derivative of $\tan(x)$ using the limit definition?
4. How does the sine addition formula help in solving trigonometric limits?
5. How do trigonometric limits differ from polynomial limits?

Tip: Understanding the sine addition formula is crucial for solving complex trigonometric limits and derivatives.