I understand the concept of how to write limits in epsilon delta style when you have a limit like this, $\lim_{x\to x_0} f(x) = f(x_0)$. But how would I write it when $\lim_{x\to \infty} f(x) = c$, for example? Would it be enough to show that $f(x)$ is bounded above or below? That is, would it be enough to show that $\exists M\in \Bbb R$ s.t. $\forall x\in \Bbb R, f(x) \lt M$ or $f(x) \gt M$? My professor hasn't really explained a whole lot about limits at $\infty$, so I am looking for outside advice. Any help will be appreciated! Thanks.
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2 Answers
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The statement $\displaystyle\lim_{x\rightarrow\infty}f(x)=c$ means for every $\varepsilon>0$ there exists an $x_0$ such that $|f(x)-c|<\varepsilon$ whenever $x>x_0$.
answered Feb 9, 2014 at 4:12
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Just being bounded certainly won't be enough to make a reasonable definition of a limit at infinity: The sine function is bounded on $\mathbb{R}$, but it keeps oscillating back and forth between $-1$ and $1$, so it's not sensible to say it has a limit at infinity.
Morally, we write
$$\lim_{x \to c} f(x) = L$$
if for all $x$ really close to $c$, $f(x)$ is really close to $L$. The same idea holds at infinity: For $x$ close to infinity (namely, really large), we want $f(x)$ close to $L$. I'll leave it to you to translate exactly what this means, more formally.