neigborhood Definition and 1 Threads

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)##?