Let be f : [a,inf)-->R is is continuous function and suppose that { limf(x) x-->inf } exist.Prove that f is boundedon [a,inf) so f is bounded function.
Let be f : [a,inf)-->R is is continuous function and suppose that { limf(x) x-->inf } exist.Prove that f is boundedon [a,inf) so f is bounded function.
Since the limit exists (say it's $\displaystyle L$) there is an integer $\displaystyle M$ such that if $\displaystyle x>M$ we get $\displaystyle \vert f(x)-L \vert <1$ and using the triangle inequality we get $\displaystyle \vert f(x) \vert < 1+\vert L \vert$. Now consider $\displaystyle f:[a,M] \rightarrow \mathbb{R}$ since it's cont. on a compact set so it attains a maximum and a minimum (say $\displaystyle A,B$ resp ) then $\displaystyle \vert f(x) \vert < \max \{ 1+\vert L \vert ,\vert A\vert ,\vert B \vert \}$ on $\displaystyle [a, \infty )$