Existence of limit question

In summary: I'm sorry, I can't remember the details. In summary, if a function is bounded both above and below but isn't monotonic and is not necessarily continuous at all points, how do you go about proving its limit exists?In summary, if a function is bounded both above and below but isn't monotonic and is not necessarily continuous at all points, you can try to find the limit using one of the methods listed above.
  • #1
DeadOriginal
274
2
Hi all,

I have a quick question about limits. This is something I should know but shame on me I forgot.

If a function is bounded both above and below but isn't monotonic and is not necessarily continuous at all points, how do I go about proving its limit exists?

In particular I am thinking about something along the lines of this function:
$$
\lim\limits_{n\rightarrow\infty}\frac{F_{n+1}}{F_{n}}
$$
where $$F_{n}$$ is the nth Fibonacci number. I know that the limit exists and I know what that limit is but say I didn't. Say I wanted to prove that it did exist. How do I do that without using the closed form expression for Fibonacci numbers? I know that it is bounded but it isn't monotonic.

This wasn't a homework question but a question that came to my mind after proving that this limit was equal to the golden ratio using the closed form expression of the Fibonacci numbers.
 
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  • #2
If a function is bounded both above and below but isn't monotonic and is not necessarily continuous at all points, how do I go about proving its limit exists?
Your question is stated in terms of functions, but your example problem is stated in terms of sequences. I'll answer for sequences.

Obviously there isn't one single method that can work in general, because the limit of such a sequence doesn't always exist.

Here are a few options to try:

1. If the differences of a sequence are alternating in sign and have limit zero, then the sequence has a limit. This is the easiest method to apply in this case. It doesn't tell you what the limit is, but you should be able to figure it out using some other method (for example, it must satisfy L = 1 + 1/L).

2. "Squeeze theorem": If you can construct an upper bound and a lower bound for the sequence such that both the upper bound and lower bound have limit L, then the sequence has limit L.

3. Prove the sequence is Cauchy.

4. Guess the limit and prove that the sequence approaches it using the definition of limit.
 
  • #3
You know that Fn+1 = Fn+Fn-1, so if the limit exists:

[tex] \lim_{n\to \infty} F_{n+1}/F_n = \lim_{n\to \infty} (F_n+F_{n-1})/F_n = \lim_{n\to \infty} 1 + F_{n-1}/F_n [/tex]

If [itex] \lim_{n\to \infty} F_{n+1}/F_n = \alpha[/itex] then the rightmost guy above is equal to [itex] 1 + 1/\alpha[/itex] so if the limit exists we get
[tex] \alpha = 1+1/\alpha[/tex]

From which you can solve for what the limit is equal to. Ironically calculating the limit is much more difficult than showing the limit exists. In this case if I remember correctly for
[tex] a_n = F_{n+1}/F_n[/tex]
the a2n form a monotonic sequence and the a2n+1 form a monotonic sequence. The argument above with some tweaking shows that the numbers they converge to are the same, and therefore the sequence [itex] a_n[/itex] converges.
 
  • #4
Office_Shredder said:
You know that Fn+1 = Fn+Fn-1, so if the limit exists:

[tex] \lim_{n\to \infty} F_{n+1}/F_n = \lim_{n\to \infty} (F_n+F_{n-1})/F_n = \lim_{n\to \infty} 1 + F_{n-1}/F_n [/tex]

If [itex] \lim_{n\to \infty} F_{n+1}/F_n = \alpha[/itex] then the rightmost guy above is equal to [itex] 1 + 1/\alpha[/itex] so if the limit exists we get
[tex] \alpha = 1+1/\alpha[/tex]

From which you can solve for what the limit is equal to. Ironically calculating the limit is much more difficult than showing the limit exists. In this case if I remember correctly for
[tex] a_n = F_{n+1}/F_n[/tex]
the a2n form a monotonic sequence and the a2n+1 form a monotonic sequence. The argument above with some tweaking shows that the numbers they converge to are the same, and therefore the sequence [itex] a_n[/itex] converges.

Yes that is exactly what I wanted to do. I have a proof that the limit is equal to the golden ratio using the closed form expression but I had to do a lot of footwork to show that the Fibonacci numbers could in fact be expressed as that closed form (I did this using induction).

I was also able to show the limit was equal to the golden ratio by using the assumption that it was equal to a. What I had forgotten was how to show functions/sequences converged if I only had boundedness. Thank you and eigenperson for reminding me.

I guess it was a bad idea to leave home without Spivak...
 
  • #5
have you thought about sin(x) as x --> infinity?
 
  • #6
That is definitely an interesting one. I have always found limits of trig functions to be kind of funky. I know that the limit doesn't exist. The only proof I know of to prove that is the epsilon delta proof that derives a contradiction.
 
  • #7
eigenperson said:
1. If the differences of a sequence are alternating in sign and have limit zero, then the sequence has a limit.

Is that true? What about the sequence [tex]S_n = (S_{n-1}+\frac{4}{n} \text{ where n is even, } S_{n-1} - \frac{2}{n-1} \text{ where n is odd}), S_0 = S_1 = 0 [/tex]Note that this generates the series ## \frac{4}{2} - \frac{2}{2} + \frac{4}{4} - \frac{2}{4} + \frac{4}{6} - \frac{2}{6} \ldots ##
 
  • #8
Yeah, I should have said "decreasing" instead of "has limit zero".
 
  • #9
eigenperson said:
Yeah, I should have said "decreasing" instead of "has limit zero".

Leibniz Criterion
 

What is the definition of a limit?

A limit is defined as the value that a function approaches as the input approaches a certain value. It is denoted by the notation "lim f(x) as x approaches a".

How do you find the limit of a function?

To find the limit of a function, you can use the limit laws, which state that the limit of a sum, difference, product, or quotient is equal to the sum, difference, product, or quotient of the limits of the individual functions. You can also use algebraic manipulation, graphs, or tables to find the limit.

What is the importance of limits in mathematics?

Limits are important in mathematics because they help us understand the behavior of a function near a certain point. They also allow us to solve problems involving rates of change, continuity, and convergence of sequences and series.

What are the common types of limits?

The three common types of limits are one-sided limits, infinite limits, and limits at infinity. One-sided limits are used when the function approaches a certain value from one direction only. Infinite limits occur when the limit of a function approaches positive or negative infinity. Limits at infinity are used to describe the behavior of a function as the input approaches infinity.

What are some applications of limits in real life?

Limits have many applications in real life, such as in physics, engineering, and economics. For example, limits are used to calculate instantaneous velocity and acceleration in physics, to design structures that can withstand maximum stress in engineering, and to model economic growth and decay.

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