# Tangent Line

Also found in: Dictionary, Acronyms, Wikipedia.

## Tangent Line

the limit of a secant. The tangent is defined as follows: Let *M* be a point on the curve *L* (Figure 1). Select a second point *M* on *L* and draw the line *MM’*. Let us consider *M* to be stationary, and let the point *M* approach *M* along the curve *L*. If, as *M* approaches *M*, the line *MM’* approaches a single definite line *MT*, then *MT* is called the tangent to the curve *L* at the point *M*. Not all continuous curves have a tangent, since the line *MM’* may not approach a limit or may approach two different limits when *M* approaches *M* from different sides of *M* (Figure 2).

The curves encountered in elementary geometry have a well-defined tangent at all points, except at certain “singular” points. If a plane curve is defined in rectangular coordinates by the equation *y* = *f*(*x*) and *f*(*x*) is differentiable at the point *x*_{0}, then the slope of the tangent at the point *M* with abscissa *x*_{0} is equal to the derivative *f*’(*x*_{0}) at the point *x*_{0}. The equation of the tangent at this point has the form

*y*−*f*(*x*_{0})=*f*’(*x*_{0})(*x*−*x*_{0})

Any line that passes through a point *M* on a surface *S* and thatlies in the tangent plane to *S* at the point *M* is called a tangentline to a surface *S* at a point *M*.