TRACON Controller experienced an airspace violation when confusing a had off with a point out and inadvertently dropping a Data Block near an airspace boundary.

Narrative

I signed on position and only about 2 minutes after finishing the briefing I took a verbal point out from the neighboring Approach. Apparently; in the process of taking the verbal point out; I dropped the tag and didn't realize I took the automated hand off on the aircraft from Center Sector X; and put an overflight tag on the target. It appeared to Sector X that I took the hand off but I was under the impression I was taking a point-out from neighboring Approach. The plane continued on course as a point out to me. After a short period of time; Sector X put the aircraft on my frequency and the pilot called and I never responded to the call. The pilot only called once and never tried again to contact me. I got very busy with other aircraft and a radio failure as well; the subject aircraft; which I believed was still a point out continued on its way and into Class B airspace. The supervisor took a phone call from the Tower to have the aircraft to contact them. I told the supervisor I took a point out and they called Sector X. The aircraft unknown to me was on my frequency and I finally shipped the aircraft to Tower. I will be extra careful about dropping tags close to airspace boundaries. Like I said; I didn't know I took the automated hand off; from Sector X; and if I did; I would never have dropped the tag and would have never taken the point out and just called RADAR contact on the aircraft.