 Solution

Use the two charts of the blowup of the affine plane at the origin. In the chart $y=xv$, the total transform isso the proper transform has equation $1-x{v}^3=0$. It misses the exceptional divisor $x=0$ and is smooth, since its derivative with respect to $x$ is $-v^3\ne 0$ there.

In the chart $x=yu$, the total transform is $y^2(u^2-y)$, so the proper transform isIts equation $u^2-y=0$ has partial derivative $-1$ with respect to $y$, so it is smooth even at its unique intersection $(u,y)=(0,0)$ with the exceptional divisor. The two chart descriptions agree on their overlap. Thus one blowup makes the cusp nonsingular, although it is tangent to the exceptional divisor there.