Question #d5f05

1 Answer
Mar 23, 2016

There isn't a precise definition of just "tangent line".

Explanation:

Definition: Line Tangent to a Circle

In Geometry a line is tangent to a circle if and only if the line and the circle have exactly one point in common. (They have one and only one point in common.)

Initial Definition: Line Tangent to a Graph of a Function

Let the point (a,f(a))(a,f(a)) lie on the graph of y=f(x)y=f(x) the line tangent to the graph at (a,f(a))(a,f(a)) is the line through that point with slope given by lim_(xrarra)(f(x)-f(a))/(x-a), provided that the limit exists.

(alternative form for the slope: lim_(hrarr0)(f(a+h)-f(a))/h)

Extension to Vertical Tangent Line to a Graph of a Function

Let the point (a,f(a)) lie on the graph of y=f(x). If lim_(xrarra)(f(x)-f(a))/(x-a) = oo or lim_(xrarra)(f(x)-f(a))/(x-a) = -oo, then the line x=a is tangent to the curve at (a,f(a)).

Other Extensions of the Definition

We can extend the definition to include "one-sided" tangent line using one-sided limits. This is useful when a function's domain is (or includes) a closed interval (or a half open interval).

We can extend the definition to cover curves that are not described by a single function, but by an equation in which one or more functions are implicit.

We can also extend the definition to the lines tangent to a surface in 3 dimensions given by an equation z=f(x,y) and variations on this idea.