Limit as x approches 0 of (2x+e^x)^1/x

Question

Limit as x approches 0

Answer 1

As x approaches zero, the contents of the brackets approaches 1, because 2x approaches 0 and e^x approaches e^0=1. The exponent 1/x approaches infinity, but 1 raised to an infinite power remains 1. Therefore the whole expression approaches 1.