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Chapter 8: Problem 31

Find the limit of the following sequences or determine that the limit does not exist. $$\left\\{n \sin \frac{6}{n}\right\\}$$

Short Answer

Expert verified
Answer: The limit does not exist.

Step by step solution

01

Find the limit of the given sequence

First, let's find the limit of the given sequence: $$\lim_{n\to\infty} n \sin \frac{6}{n}$$
02

Use properties of sine function

As we know that the sine function is always bounded in the interval \([-1,1]\), that is, \(-1 \leq \sin(x) \leq 1\) for all \(x\). So, we have: $$-1\leq \sin \frac{6}{n} \leq 1$$
03

Introduce the variable n and apply Squeeze Theorem indirectly

We can multiply the inequality by the value of n, because it is positive and we don't need to change the inequality direction: $$-n \leq n\sin \frac{6}{n} \leq n$$ Now, we can apply the limit to each part of the inequality: $$\lim_{n\to\infty} -n \leq \lim_{n\to\infty} n\sin \frac{6}{n} \leq \lim_{n\to\infty} n$$
04

Calculate the limits separately

Now we calculate the limits separately: $$\lim_{n\to\infty} -n = -\infty$$ $$\lim_{n\to\infty} n = \infty$$ By observing the behavior of the inequality, we can see that the middle limit does not have a definite value and is being squeezed between negative and positive infinity.
05

Conclude the solution

Since the limit of the sequence behaves as such, we can conclude that: $$\lim_{n\to\infty} n \sin \frac{6}{n}$$ does not exist.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Squeeze Theorem
The Squeeze Theorem, also known as the Sandwich Theorem, is a vital concept in calculus, particularly useful when dealing with limits. Imagine you have two functions that 'sandwich' another function between them, and these two functions have the same limit at a certain point. The Squeeze Theorem tells us that the function in the middle must also approach the same limit at that point.

For example, if you have functions f(x), g(x), and h(x), with f(x) 鈮� g(x) 鈮� h(x) for all x in some interval, and if the limit of both f(x) and h(x) as x approaches a certain value is L, then the limit of g(x) as x approaches that value must also be L. This theorem is particularly useful for dealing with tricky limits where direct computation isn't straightforward.
Properties of Sine Function
The sine function is a fundamental trigonometric function that exhibits several important properties which are crucial in the study of sequences and limits. One key property is that the sine function is always bounded. Specifically, for any real number x, the value of sine x lies within the closed interval \[ -1, 1 \], meaning \( -1 \leq \sin(x) \leq 1 \).

This property of being bounded is what makes the sine function well-behaved in mathematical analysis. No matter the input, the output never exceeds these bounds, which can be remarkably useful when combining the sine function with other variables, especially when we aim to understand the behavior of these functions as variables grow large or small.
Bounded Functions
A bounded function is one for which the absolute value of the function is limited to a certain range for all inputs in its domain. This means that the output of the function does not exceed a certain positive value, nor falls below a certain negative value.

Examples of Bounded Functions

  • The sine and cosine functions are classic examples of bounded functions, with outputs ranging from -1 to 1.
  • Constant functions, where the output is the same no matter the input, are also bounded.
Understanding bounded functions is important in calculus because they offer predictability and constraints that help in analyzing the behavior of sequences and functions, especially when approaching limits.
Infinite Limits
In infinite limits, we explore the behavior of functions as they increase or decrease without bound. In other words, as the input grows very large or very small, we try to describe the behavior of the function's output. These limits are not finite real numbers, but rather 'infinities'.

Understanding infinite limits is crucial for determining how a function behaves at its vertical asymptotes, where the function might shoot up to positive infinity (\( +\infty \)) or plummet to negative infinity (\( -\infty \)). In the context of sequences, an infinite limit means that the terms of the sequence grow larger and larger without settling down to a finite number. This understanding can help us grasp concepts related to growth rates and the 'end behavior' of functions as we venture towards the edges of our mathematical landscape.

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Most popular questions from this chapter

Here is a fascinating (unsolved) problem known as the hailstone problem (or the Ulam Conjecture or the Collatz Conjecture). It involves sequences in two different ways. First, choose a positive integer \(N\) and call it \(a_{0} .\) This is the seed of a sequence. The rest of the sequence is generated as follows: For \(n=0,1,2, \ldots\) $$a_{n+1}=\left\\{\begin{array}{ll} a_{n} / 2 & \text { if } a_{n} \text { is even } \\ 3 a_{n}+1 & \text { if } a_{n} \text { is odd .} \end{array}\right.$$ However, if \(a_{n}=1\) for any \(n,\) then the sequence terminates. a. Compute the sequence that results from the seeds \(N=2,3\), \(4, \ldots, 10 .\) You should verify that in all these cases, the sequence eventually terminates. The hailstone conjecture (still unproved) states that for all positive integers \(N\), the sequence terminates after a finite number of terms. b. Now define the hailstone sequence \(\left\\{H_{k}\right\\},\) which is the number of terms needed for the sequence \(\left\\{a_{n}\right\\}\) to terminate starting with a seed of \(k\). Verify that \(H_{2}=1, H_{3}=7\), and \(H_{4}=2\). c. Plot as many terms of the hailstone sequence as is feasible. How did the sequence get its name? Does the conjecture appear to be true?

Consider the number \(0.555555 \ldots,\) which can be viewed as the series \(5 \sum_{k=1}^{\infty} 10^{-k} .\) Evaluate the geometric series to obtain a rational value of \(0.555555 .\) b. Consider the number \(0.54545454 \ldots\), which can be represented by the series \(54 \sum_{k=1}^{\infty} 10^{-2 k} .\) Evaluate the geometric series to obtain a rational value of the number. c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length \(p,\) say, \(n_{1}, n_{2} \ldots ., n_{p},\) where \(n_{1}, \ldots, n_{p}\) are integers between 0 and \(9 .\) Explain how to use geometric series to obtain a rational form for \(0 . \overline{n_{1}} n_{2} \cdots n_{p}\) d. Try the method of part (c) on the number \(0 . \overline{123456789}=0.123456789123456789 \ldots\) e. Prove that \(0 . \overline{9}=1\)

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{7^{n}}{n^{7} 5^{n}}$$

A tank is filled with 100 L of a \(40 \%\) alcohol solution (by volume). You repeatedly perform the following operation: Remove 2 L of the solution from the tank and replace them with 2 L of \(10 \%\) alcohol solution. a. Let \(C_{n}\) be the concentration of the solution in the tank after the \(n\) th replacement, where \(C_{0}=40 \% .\) Write the first five terms of the sequence \(\left\\{C_{n}\right\\}\). b. After how many replacements does the alcohol concentration reach \(15 \% ?\). c. Determine the limiting (steady-state) concentration of the solution that is approached after many replacements.

Use the formal definition of the limit of a sequence to prove the following limits. $$\lim _{n \rightarrow \infty} \frac{3 n^{2}}{4 n^{2}+1}=\frac{3}{4}$$
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