 This includes cases where one slope approaches positive infinity while the other approaches negative infinity, leading to an infinite jump discontinuity

When the curve is given by "y" = "f"("x") then the slope of the tangent is formula_7
so by the point–slope formula the equation of the tangent line at ("X", "Y") is
where ("x", "y") are the coordinates of any point on the tangent line, and where the derivative is evaluated at formula_9.When the curve is given by "y" = "f"("x"), the tangent line's equation can also be found by using polynomial division to divide formula_10 by formula_11; if the remainder is denoted by formula_12, then the equation of the tangent line is given by

When the equation of the curve is given in the form "f"("x", "y") = 0 then the value of the slope can be found by implicit differentiation, giving
The equation of the tangent line at a point ("X","Y") such that "f"("X","Y") = 0 is then
This equation remains true if formula_16 but formula_17 (in this case the slope of the tangent is infinite).If formula_18 the tangent line is not defined and the point ("X","Y") is said singular.