Proving Boundedness of Continuous Functions in [a,+∞] with Limits

[sedaw]

need to prove that f(x) bounded if f(x) continuous in [a,+infinite] and if there's a limit while x goes to +infinite.


I would really appreciate any kind of help !

 

[HallsofIvy]

need to prove that f(x) bounded if f(x) continuous in [a,+infinite] and if there's a limit while x goes to +infinite.


I would really appreciate any kind of help !

I assume you mean prove that f is bounded in [a, infinity). Otherwise, it is not true. Let the limit be L. By definition of limit at infinity, that means that there exist some R such that if x> R, |f(x)- L|< 1 so for x> R, L-1< f(x)< L+1. Further since f(x) is continuous, f is bounded on the close, bounded interval [0, R]. Put those two together.

 

[sedaw]

hello HallsofIvy ! , " I assume you mean prove that f is bounded in [a, infinity)."

that is right , i don't understand why did u choose epsilon=1 is it necessary ?

TNX!

 

[HallsofIvy]

Since the problem is only to prove that f is bounded, you can choose $\epsilon$ to be any (non-zero) number. "1" happened to be convenient.

If |f(x)|< B on [0, R] and |f(x)|< 1 on [R, infinity), what is a bound on f?