See image 1. Circles A,B,C and D contain the anchor points of the blue shape. Let's assume you want to continue straight line segments AB and DC to form a sharp corner.
Draw lines AB and DC. To do it exactly have smart guides and snap to points ON, no other snaps!
Holding Shift at the same time drag the new line objects with the normal selection tool to bigger sizes to make a crossing
With the anchor removal tool remove anchor B and with the anchor type conversion tool make C a corner point
Drag C to the crossing of the lines with the direct selection tool
NOTE: This receipe is useless, if you literally want to find the point where the extensions of the handles meet, like in this:
Blue shape can be curved in both sides of B and C, but have tangents, at least one sided tangents. Finding those tangents need something smarter - for example a third party plugin. Check, if Astute Graphics has a good one. I haven't one.
Alternatively you can goto Inkscape. There's a path visualization extension "Draw handles" which really inserts to the image all handles as lines. It's no problem to extend them:
Then you can draw the exact needed add-on shape and make an union: