L'Hospital's Rule: Zoom In Until 0/0 Becomes Two Straight Lines
f(x) = sin(pi(x - 1)) and g(x) = ln x both hit zero at x = 1, so f(x)/g(x) there is 0/0.
Zoom in on that point. The closer you look, the straighter both curves get, until they are just their tangent lines: f rises pi for every step dx, g rises 1. Over the same dx the two heights are df(a) = pi dx and dg(a) = 1 dx, and dx cancels:
lim f(x)/g(x) = lim (df/dx)/(dg/dx) = lim f'(x)/g'(x) = pi / 1 = pi
The live ratio f(x)/g(x) closes in on 3.1416 as x slides toward 1 from both sides.
When b² − 4ac < 0, the parabola never touches the x-axis. But you can still draw its roots.
Rotate it 180° about the vertex. The flipped curve crosses the axis at (−b ± √(4ac − b²))/2a.
Turn those two points 90° about their midpoint (−b/2a, 0). Treat the plane as complex, and they land on (−b ± i√(4ac − b²))/2a.
Complex roots aren't missing. They're just off the real line.