https://file.aiquickdraw.com/mathbot/file/1723506781371tz20k1za.jpg

Let's solve the limits step by step.

### Problem (a):
\[
\lim_{x \to \infty} \left( 2 \sin \frac{1}{x} - \cos \frac{1}{2x} \right)
\]

**Solution:**

1. **Evaluate each trigonometric function as \(x \to \infty\):**
   - \(\sin \frac{1}{x} \to \sin 0 = 0\)
   - \(\cos \frac{1}{2x} \to \cos 0 = 1\)

2. Substitute these values into the limit expression:
   \[
   \lim_{x \to \infty} \left( 2 \sin \frac{1}{x} - \cos \frac{1}{2x} \right) = 2 \cdot 0 - 1 = -1
   \]

### Answer: \( \boxed{-1} \)

---

### Problem (b):
\[
\lim_{x \to \infty} \left( \tan \frac{1}{3x} + x \sin^4 \frac{1}{x} \right)
\]

**Solution:**

1. **Evaluate each function as \(x \to \infty\):**
   - \(\tan \frac{1}{3x} \to \tan 0 = 0\)
   - For the second term, consider \( \sin^4 \frac{1}{x} \):
     \[
     \sin^4 \frac{1}{x} \approx \left(\frac{1}{x}\right)^4 \text{ for large } x
     \]
     Then, \( x \sin^4 \frac{1}{x} \approx x \left(\frac{1}{x}\right)^4 = \frac{1}{x^3} \to 0 \text{ as } x \to \infty \).

2. Substitute these values into the limit expression:
   \[
   \lim_{x \to \infty} \left( \tan \frac{1}{3x} + x \sin^4 \frac{1}{x} \right) = 0 + 0 = 0
   \]

### Answer: \( \boxed{0} \)

---

### Problem (c):
\[
\lim_{x \to \infty} \left( \frac{\sin \frac{2}{3x}}{\frac{3}{2x}} + \frac{\sin \frac{1}{3x}}{\tan \frac{3}{2x}} \right)
\]

**Solution:**

1. **Evaluate each expression separately:**

   - For the first term:
     \[
     \lim_{x \to \infty} \frac{\sin \frac{2}{3x}}{\frac{3}{2x}} \text{ can be simplified using the small-angle approximation } \sin \theta \approx \theta:
     \]
     \[
     \frac{\sin \frac{2}{3x}}{\frac{3}{2x}} \approx \frac{\frac{2}{3x}}{\frac{3}{2x}} = \frac{4}{9}
     \]

   - For the second term:
     \[
     \lim_{x \to \infty} \frac{\sin \frac{1}{3x}}{\tan \frac{3}{2x}} \text{ also simplifies similarly } \sin \theta \approx \theta \text{ and } \tan \theta \approx \theta:
     \]
     \[
     \frac{\frac{1}{3x}}{\frac{3}{2x}} = \frac{2}{9}
     \]

2. Adding these results:
   \[
   \lim_{x \to \infty} \left( \frac{4}{9} + \frac{2}{9} \right) = \frac{6}{9} = \frac{2}{3}
   \]

### Answer: \( \boxed{\frac{2}{3}} \)

---

Would you like more details on any of these solutions? Here are 8 related questions you might find interesting:

1. How do small-angle approximations work in limit problems?
2. What is L'Hôpital's rule and when is it applicable?
3. How can trigonometric limits be evaluated when \(x \to 0\) or \(x \to \infty\)?
4. How does the squeeze theorem help in solving limits?
5. What happens when both the numerator and denominator approach zero in a limit?
6. Can we generalize these limits for higher-order sine and cosine functions?
7. How do asymptotic expansions assist in evaluating limits?
8. What are the key differences between evaluating limits at infinity versus finite points?

**Tip:** When dealing with trigonometric limits, always consider if small-angle approximations or series expansions might simplify the problem.

Explore how to solve limit problems involving trigonometric functions step-by-step. Learn about evaluating limits as x approaches infinity using small-angle approximations. Suitable for advanced high school and college 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