Question

Question #d5f05

Answer 1

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))` lie on the graph of `y=f(x)` the line tangent to the graph at `(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.