The limit in question is:

\[
\lim_{h \to 0} \frac{\sin(x + h) + \sin(x)}{h}
\]

To evaluate this limit, let's rewrite the expression using the sum of sines formula. The sum of sines identity is:

\[
\sin(a) + \sin(b) = 2 \sin\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right)
\]

However, in this case, it’s more straightforward to work with the definition of derivatives. The expression looks like the derivative of \( \sin(x) \).

Recall that the derivative of \( \sin(x) \) is:

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

So the expression \(\frac{\sin(x + h) - \sin(x)}{h}\) as \(h \to 0\) gives us the derivative of \( \sin(x) \) at \(x\), which is:

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

But in the problem, you have a **plus** sign between the sines instead of a minus. Therefore, this expression does not correspond directly to the derivative.

Thus, the original limit:

\[
\lim_{h \to 0} \frac{\sin(x + h) + \sin(x)}{h}
\]

**does not exist**, since the sum of sines over \(h\) does not yield a finite value when \(h\) approaches 0.

### Do you have any questions or need more details?

---

Here are 5 related questions to expand on this topic:
1. What is the derivative of the function \( \cos(x) \)?
2. How would you approach the limit of \(\frac{\cos(x+h) - \cos(x)}{h}\) as \(h \to 0\)?
3. Can you explain the small angle approximation for \(\sin(h)\) when \(h\) is near 0?
4. How would you calculate \(\lim_{h \to 0} \frac{\sin(x + h) - \sin(x - h)}{2h}\)?
5. What is the geometric interpretation of the derivative of a sine function?

**Tip:** When working with limits involving trigonometric functions, recognizing patterns that resemble derivatives can simplify the solution.

Explore the limit lim h->0 (sin(x+h) + sin(x))/h, understand why it does not exist, and learn key concepts in trigonometry and limits. This problem uses the sum of sines identity and the limit definition of derivatives, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation