Is a Real funcion with a Limit Bounded?

[jacksonjs20]

Hi, just a quick question.

Let f be real function s.t. the limit of f as x approaches a equals L.

Is f bounded?
i.e. is it sufficient to assume a function is bounded if it has a limit.

Thanks to all who may reply.

 

[pessimist]

Yes it is. Use the definition of limit and choose your epsilon to be a certain constant (say 1). Then work your magic.

 

[alexfloo]

Rather, it is bounded on some open set containing a.

It might go off to infinity elsewhere; you just know that it doesn't go off to infinity right at that point.

 

[gb7nash]

Let f be real function s.t. the limit of f as x approaches a equals L.

Is f bounded?

There's not enough information to tell. The only x value that you can say f is bounded is x = a. For every other x value, we have no idea if there exists an x asymptote. Like the previous poster suggested, we only know that it's bounded on some open interval containing a.

 

[SteveL27]

Hi, just a quick question.

Let f be real function s.t. the limit of f as x approaches a equals L.

Is f bounded?
i.e. is it sufficient to assume a function is bounded if it has a limit.

Thanks to all who may reply.

Just take f(x) = x. The limit of f as x -> a is defined and finite for every a, but f is not bounded.

 

[gb7nash]

I noticed a mistake in my previous post. f(x) isn't guaranteed to even exist for x = a. However, f(x) is still bounded in some neighborhood of x = a, since the limit exists there.