Let's analyze each sequence one by one:

(a) $\left\{ \frac{\sin(5n)}{n} \right\}$

• The function $\sin(5n)$ is bounded, meaning that for any $n$, $-1 \leq \sin(5n) \leq 1$.
• As $n \to \infty$, the denominator $n$ grows without bound while $\sin(5n)$ oscillates between -1 and 1.
• Therefore, $\frac{\sin(5n)}{n} \to 0$ as $n \to \infty$.

Limit: 0

(b) $\left\{ \frac{\sin(4n)}{n} \right\}$

• Similarly, $\sin(4n)$ is bounded by -1 and 1.
• As $n \to \infty$, the denominator $n$ increases without bound while $\sin(4n)$ oscillates.
• Hence, $\frac{\sin(4n)}{n} \to 0$ as $n \to \infty$.

Limit: 0

(c) $\left\{ \sin\left( \frac{\pi(6n + 1)}{2} \right) \right\}$

• First, simplify the expression inside the sine function: $\sin\left( \frac{\pi(6n + 1)}{2} \right) = \sin\left( 3n\pi + \frac{\pi}{2} \right)$ Using the periodicity of sine, $\sin(x + 2k\pi) = \sin(x)$, we get: $\sin\left( 3n\pi + \frac{\pi}{2} \right) = \sin\left( \frac{\pi}{2} \right) = 1$
• Therefore, this sequence simplifies to $1$ for all $n$.

Limit: 1

(d) $\left\{ n \sin(6\pi n) \right\}$

• The term $6\pi n$ is a multiple of $2\pi$, and we know that $\sin(k \cdot 2\pi) = 0$ for any integer $k$.
• Thus, $\sin(6\pi n) = 0$ for all $n$.

Therefore, $n \sin(6\pi n) = n \cdot 0 = 0$.

Limit: 0

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Summary of Limits:

• (a) $\lim_{n \to \infty} \frac{\sin(5n)}{n} = 0$
• (b) $\lim_{n \to \infty} \frac{\sin(4n)}{n} = 0$
• (c) $\lim_{n \to \infty} \sin\left( \frac{\pi(6n + 1)}{2} \right) = 1$
• (d) $\lim_{n \to \infty} n \sin(6\pi n) = 0$

Would you like more details on any specific sequence?

Relative Questions:

1. What is the general method to calculate limits for trigonometric sequences?
2. Why do sine functions oscillate, and how does this affect limits?
3. How does periodicity of trigonometric functions impact the convergence of sequences?
4. What are common types of divergent sequences in trigonometry?
5. How does L'Hopital's Rule apply to limit problems involving trigonometric functions?

Tip:

Always check for bounded behavior in oscillating functions like sine or cosine to determine the convergence behavior of sequences.