Let f be a function from real numbers to real numbers and let f(x) be a continous function, such that:
$$\lim_{x\rightarrow\infty}f(x)$$ $$\lim_{x\rightarrow-\infty}f(x)$$
are bounded.
How do I prove that f(x) is bounded?
Proving that function is bounded when its continous and its limits at infinity are bounded.
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$\begingroup$
limits
continuity
epsilon-delta
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0It might be worth looking at $f([-1,1])$, then $f([-2,2])$, ... then $f([-n,n])$. This is a continuous function on a compact set, so for any finite $n$ you have that $f$ is bounded. Then you just need to find a way to incorporate the behavior at infinity. – 2017-02-28
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0Use the definition of limit to obtain a bound for large $x$. Then reduce to the case of a compact interval of definition – 2017-02-28
1 Answers
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Let $L_+$ and $L_-$ be the two limits at $\infty$ and $-\infty$.
There exist $M_+$ and $M_-$ such that $|f(x)-L_+| < 1$ for all $x > M_+$, and $|f(x) - L_-|<1$ for all $x < M_-$. Then all you need to do is bound $f$ on the compact interval $[M_-,M_+]$.