I got this question for homework, and I am not sure I fully understand it:
Prove that the next two statements are equivalent: $\exists L\forall\epsilon\gt0\exists P:P\lt x\rightarrow\|f(x)-L\|\lt \epsilon$ $\exists L\forall\epsilon\gt0\exists P\gt0:P\lt x\rightarrow\|f(x)-L\|\lt \epsilon$
Both statements say that $f(x)$ has a limit $L$ when $x\rightarrow\infty$. The definition of limit in infinity does not require anything from $P$, so what am I supposed to show here? Thanks