How do I blend out to white?

[enginestar]

Can someone tell me how I blend out to white?

Problem: I've learned how to adjust colour levels and make the background become white.
This is perfect... except most of the time, around the edges... there are bits of grey. These show when the placed on a pure white background.

Is there some way of making sure the edges of the background is pure white?
I was thinking of a gradient blend somehow?

ELSE... am I doing something wrong in the first place...?

Thanks.