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: $\boxed{-1}$

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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: $\boxed{0}$

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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: $\boxed{\frac{2}{3}}$

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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.