$\lim_{ t\rightarrow \infty} h(t)= \infty$

Question

If $f : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function. What is the definition of what it means to say $\lim_{ t\rightarrow \infty} f(t)= \infty$?

Would this be the correct definition? The function $f$ has a limit at $\infty$ if there exists a number $L \in \mathbb{R}$ such that given $\epsilon > 0$, there exists a real number $M$ for which $|f(x)-L|< \epsilon$ for all $x \in$ Dom$ f \cap (M, \infty)$.

Answer 1

The condition that $|f(x) - L|<\varepsilon$ would mean that as $\varepsilon \to 0$ then $f(x)\to L$. Intuitively what you mean when you say $$\lim_{x\to\infty}f(x) = \infty$$ is that no matter how large a number $N$ you pick, there will always be a point $x$ after which your function $f(x)$ will be greater than $N$. Formally we can write the following: For all $N>0$, there exists an $x'>0$ such that for all $x>x'$ we have $f(x) > N$.

Answer 2

I think the following definition works, and $f$ need not be continuous.

For all $L$ there exists $M$ such that $f(x)>L$ for all $x>M$.