Can a Non-Decreasing Function Have a Limit at Infinity?

In summary, if f : R -> R is a non-decreasing function, then the limit as x tends to infinity of f(x) can either be infinity, minus infinity, or a real number. This can be proven by using the least upper bound property and an epsilon-N argument, or by showing that the function is either bounded or unbounded. If the function is not strictly increasing, but is still non-decreasing, the limit may still exist but cannot be proven without further information.
  • #1
Legendre
62
0
I read that "if f : R -> R is an increasing function, then limit as x tend to infinity of f(x) is either infinity, minus infinity or a real number". f an increasing function means { x < y } => { f(x) < or = f(y) }.

How do I prove this (if it is true)? Can I apply this to a function g : R -> [0,1]?

P.S.
I am not looking for a precise proof. A loose discussion of the way to prove this would be fine.

Thanks!
 
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  • #2
Actually, such a function cannot have a limit of negative infinity. If the function is bounded (there exist a number M such that f(x)< M for all x) then f(x) converges to a real number. If it is unbounded, then it goes to positive infinity.

How you would prove that depends on what you have available. Can you use the "least upper bound property"- that every nonempty set of real numbers with an upper bound has a least upper bound? If so then: Since f(x)< M for all x, the set {f(x)} has M as upper bound and so has a least upper bound, [itex]\alpha[/itex]. You can use an "[itex]\epsilon[/itex], N" argument to show that [itex]\alpha[/itex] is the limit.

If the function is not bounded, then, given any Y> 0, there exist x0 such that f(x0)> Y. But then if x1> x0, because f is increasing, f(x1)> f(x0)> Y.
 
  • #3
What other limits can there possibly be, other than infinity, minus infinity, or a real number?

The function obviously is from the real plane to the real plane...so there can be no other limits.

Are you perhaps asking to prove that such a limit exists?
 
  • #4
There can be no limit at all.

For example sin(x) has no limit at infinity (proven simply with Heine Theorem)

As also xsin(x), although it seems it goes to infinity, it will actually oscillate between very large negative numbers and very large positive numbers (to prove this, Heine's Theorem won't work, and the resort is the original definition of the limit)
 
  • #5
HallsofIvy said:
Actually, such a function cannot have a limit of negative infinity. If the function is bounded (there exist a number M such that f(x)< M for all x) then f(x) converges to a real number. If it is unbounded, then it goes to positive infinity.

Great idea. However, I am terribly sorry if I didn't make it clear that what I meant by "increasing" is not "strictly increasing". Perhaps I should have mention that f is non-decreasing in the first place!

The case that I have trouble with is when f is NOT strictly increasing and yet IS non-decreasing.
 

FAQ: Can a Non-Decreasing Function Have a Limit at Infinity?

What is the definition of a limit of a function at infinity?

The limit of a function at infinity is the value that the function approaches as the input variable (usually denoted as x) grows without bound. It is also known as the limit at infinity or the asymptotic limit.

How do you determine the limit of a function at infinity?

To determine the limit of a function at infinity, one can use the following rules:

  • If the function is a polynomial, the limit at infinity is either positive infinity, negative infinity, or the leading coefficient divided by the leading power of x.
  • If the function contains a fraction, the limit at infinity can be found by dividing the leading coefficient of the numerator by the leading coefficient of the denominator.
  • If the function contains radicals, the limit at infinity can be found by dividing the leading coefficient of the radicand by the leading coefficient of the root.
  • If the function contains exponential or logarithmic functions, the limit at infinity can be found by using L'Hopital's Rule or by factoring out the highest power of x in the exponent.

What are the possible types of limits at infinity?

There are three possible types of limits at infinity:

  • The limit is equal to a finite number.
  • The limit is equal to positive infinity.
  • The limit is equal to negative infinity.

Can a function have more than one limit at infinity?

Yes, a function can have more than one limit at infinity. This can occur when the function has different behaviors as x approaches positive and negative infinity, or when the function has discontinuities at different values of x.

How are limits at infinity related to the horizontal asymptote of a function?

Limits at infinity and horizontal asymptotes are closely related. If the limit at infinity of a function is a finite number, then the function will have a horizontal asymptote at that value. However, if the limit at infinity is positive or negative infinity, then the function will not have a horizontal asymptote.

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