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Chapter 9: Problem 54

Find the limit of the following sequences or state that they diverge. $$\left\\{\frac{\sin 6 n}{5 n}\right\\}$$

Short Answer

Expert verified
**Question:** Determine the limit of the sequence given by the function \(a_n = \frac{\sin 6n}{5n}\). **Answer:** The limit of the given sequence is 0.

Step by step solution

01

The sequence function is given as \(a_n = \frac{\sin 6n}{5n}\). #Step 2: Find the limit of the sequence as n approaches infinity (if it exists)#

We are looking for \(\lim_{n\rightarrow\infty} \frac{\sin 6n}{5n}\). #Step 3: Rewrite the limit using substitution#
02

Let \(x = 6n\), so that when \(n \rightarrow \infty\), \(x \rightarrow \infty\). Now, the limit becomes \(\lim_{x\rightarrow\infty} \frac{\sin x}{\frac{5}{6}x}\). #Step 4: Split the limit into two parts#

The limit can be rewritten as \(\frac{1}{\frac{5}{6}}\lim_{x\rightarrow\infty} \frac{\sin x}{x}\). #Step 5: Evaluate the limit of the \(\frac{\sin x}{x}\) part#
03

We cannot directly apply the standard limit property \(\lim_{x\rightarrow 0} \frac{\sin(x)}{x}=1\) to \(\lim_{x\rightarrow\infty} \frac{\sin x}{x}\) since x is approaching infinity, not zero. However, we can analyze its behavior as x approaches infinity. The sine function oscillates between -1 and 1, while the denominator keeps increasing infinitely. That means the fraction \(\frac{\sin x}{x}\) will constantly shrink as x gets larger. This pushes the limit towards zero: \(\lim_{x\rightarrow\infty} \frac{\sin x}{x} = 0\). #Step 6: Evaluate the overall limit#

Now, we can plug this result back into the overall limit. The limit of the given sequence is \(\frac{1}{\frac{5}{6}}(0) = 0\). Therefore, the limit of the sequence is 0.

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Most popular questions from this chapter

Showing that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} \operatorname{In} 1734,\) Leonhard Euler informally proved that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} .\) An elegant proof is outlined here that uses the inequality $$ \cot ^{2} x<\frac{1}{x^{2}}<1+\cot ^{2} x\left(\text { provided that } 0

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