The function f:R∼{0}→R    f(x)=1x−2e2x−1  can be made continuous at x=0  by defining f(0)  as:

f(0)=RHL          =limx→0+f(x)=limh→0f(0+h)          =limh→0(1h−2e2h−1)          =limh→0(e2h−1−2h)h(e2h−1)2h.2h          =limh→0e2h−1−2h2h2limh→0(2he2h−1)         =limh→0(1+2h+(2h)22!+...)−1−2h2h2.1         =1