- #1
Lotto
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- TL;DR Summary
- I was given this definiton of a limit:
Let us have a function ##f: \mathbb{R} \rightarrow \mathbb{C}##. Let ##x_0 \in \mathbb{R}^*## and ##L \in \mathbb{R}^*##. We say ##L## is a limit of a function ##f## for ##x## goes to ##x_0## if
##\forall \varepsilon >0 \, \exists \delta >0 \, \forall x \in P_{\delta}(x_0): f(x) \in U_{\varepsilon}(L)##
(##P## and ##U## are neighborhoods)
Is this definition valid for every type of limit?
I suppose that it is because we are in extended real numbers. But the definition of a limit when ##x_0 = \infty## and let's say ##L=\infty## is different. Why are these definitions equivalent? Isn't the key that ##U_{\varepsilon}(\infty)=\left(\frac {1}{\varepsilon},\infty\right)##?