Math Problem Statement

Kerjakan soal tersebut secara detail

Hasil dari limit lim (x -> 0) 4x * tan(3x) / (1 - cos(4x)) adalah?

Solution

To solve this limit problem:

limx04xtan(3x)1cos(4x)\lim_{x \to 0} \frac{4x \cdot \tan(3x)}{1 - \cos(4x)}

we'll use limit properties and trigonometric approximations for small xx. Here is the detailed solution:

Step 1: Expand Trigonometric Functions

For small values of xx, we use the following approximations:

  • tan(3x)3x\tan(3x) \approx 3x
  • 1cos(4x)(4x)22=8x21 - \cos(4x) \approx \frac{(4x)^2}{2} = 8x^2

Substitute these approximations into the limit:

limx04x3x8x2\lim_{x \to 0} \frac{4x \cdot 3x}{8x^2}

Step 2: Simplify the Expression

Simplify the above expression:

limx012x28x2\lim_{x \to 0} \frac{12x^2}{8x^2}

The x2x^2 terms cancel out:

limx0128=32\lim_{x \to 0} \frac{12}{8} = \frac{3}{2}

Final Answer

The result of the limit is 32\frac{3}{2}, which corresponds to option (c).

Conclusion

Thus, the correct answer is:

c. 32\frac{3}{2}

Would you like further details or have any other questions?


Here are 5 related questions that expand on the topic:

  1. How do you derive the small-angle approximations for trigonometric functions?
  2. What other methods can be used to solve limits involving trigonometric functions?
  3. How can L'Hôpital's rule be applied to solve this type of limit?
  4. Why does tan(x)\tan(x) approximate to xx as x0x \to 0?
  5. What are some common pitfalls when solving trigonometric limits?

Tip: When solving limits involving trigonometric functions, remember that small-angle approximations can simplify the calculations significantly, especially when xx approaches 0.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Limits
Trigonometry
Small-angle approximations

Formulas

lim (x -> 0) (tan(x) ≈ x)
lim (x -> 0) (1 - cos(x) ≈ x^2 / 2)

Theorems

Limit properties
Small-angle approximations

Suitable Grade Level

Grade 12 or early college