I was reading a rather old textbook on Lie groups. He gave us an infinitesimal transformation:
$$ L = 2 x \frac{d}{dx} + 3 y \frac{d}{dy} $$
And he asked what was the global transformation generated by this.
$$e^{ t( 2 x \frac{d}{dx} + 3 y \frac{d}{dy})} f(x,y)= \text{?} $$
I could do a much easier one:
$$e^{ t \frac{d}{dx} } f(x)= f(x) + t f'(x) + (t^2/2)f''(x) +\dots= f(x+t)$$ This is just Taylor's theorem.