https://file.aiquickdraw.com/mathbot/file/1726445081367ebnad6gz.jpg

The problem in the image asks to estimate the limit of the following expression:

\[
\lim_{h \to 0} \frac{3\cos(h) - 3}{h}
\]

### Step-by-step Solution:

1. **Simplify the expression:**

   The expression can be rewritten as:

   \[
   \lim_{h \to 0} \frac{3[\cos(h) - 1]}{h}
   \]

   This is now a well-known limit problem where \(\lim_{h \to 0} \frac{\cos(h) - 1}{h}\) is evaluated.

2. **Known Limit Formula:**

   From trigonometric limit rules, we know that:

   \[
   \lim_{h \to 0} \frac{\cos(h) - 1}{h} = 0
   \]

3. **Apply the result to the original expression:**

   Using the fact that \(\lim_{h \to 0} \frac{\cos(h) - 1}{h} = 0\), we multiply the result by 3:

   \[
   \lim_{h \to 0} \frac{3[\cos(h) - 1]}{h} = 3 \times 0 = 0
   \]

Thus, the value of the limit is \( \boxed{0} \).

### Final Answer:
The answer to the limit is 0.

---

Would you like more details or have any questions? Here are some related questions to explore:

1. What are other common trigonometric limits, and how are they derived?
2. How can L'Hopital's rule be applied to solve limits like this one?
3. What is the geometric interpretation of limits involving trigonometric functions?
4. How does the small angle approximation for cosine help in calculating limits?
5. How do limits apply to the derivative definition for trigonometric functions?

**Tip:** When solving trigonometric limits, recognizing well-known limit forms can simplify your calculations!

Estimate the limit of lim(h->0) (3cos(h) - 3) / h.

Learn how to estimate the limit of lim(h->0) (3cos(h) - 3) / h using known trigonometric limit properties. This problem is suited for high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation