How can I show that if P = (x, 0) is a point on an elliptic curve, then 2P = inf ?
Try to think of this geometrically.
Say you have a typical elliptic curve over $\displaystyle \mathbb{R}$.
Now visualize it.
By $\displaystyle (x,0)$ you mean the point with cross the x-axis.
The thing is when we add the point to itself we need to draw the tangent line and find intersection point with the line on the curve (which is then reflected through the x-axis by Weierstrass group law definitions).
However, this tangent line is vertical and it never cross over.
Thus, this is the point at infinity, $\displaystyle (x,0)+(x,0) = \infty$.