The formulas above fail when the point is a singular point. In this case there may be two or more branches of the curve that pass through the point, each branch having its own tangent line. When the point is the origin, the equations of these lines can be found for algebraic curves by factoring the equation formed by eliminating all but the lowest degree terms from the original equation. Since any point can be made the origin by a change of variables (or by translating the curve) this gives a method for finding the tangent lines at any singular point.

For example, the equation of the limaçon trisectrix shown to the right is

Expanding this and eliminating all but terms of degree 2 gives

which, when factored, becomes

So these are the equations of the two tangent lines through the origin.

When the curve is not self-crossing, the tangent at a reference point may still not be uniquely defined because the curve is not differentiable at that point although it is differentiable elsewhere. In this case the left and right derivatives are defined as the limits of the derivative as the point at which it is evaluated approaches the reference point from respectively the left (lower values) or the right (higher values). For example, the curve "y" = |"x" | is not differentiable at "x" = 0: its left and right derivatives have respective slopes –1 and 1; the tangents at that point with those slopes are called the left and right tangents.

Sometimes the slopes of the left and right tangent lines are equal, so the tangent lines coincide. This is true, for example, for the curve "y" = "x" , for which both the left and right derivatives at "x" = 0 are infinite; both the left and right tangent lines have equation "x" = 0.

Multiple tangents at a point
Intuitive description
Tangent line to a curve
Analytical approach