tangent line

If the curve   $y=f(x)$  of $xy$-plane is sufficiently smooth in its point  $(x_0,y_0)$  and in a neighborhood of this, the curve may have a tangent line (or simply ) in  $(x_0,y_0)$.  Then the tangent line of the curve  $y=f(x)$  in the point  $(x_0,y_0)$  is the limit position of the secant line through the two points  $(x_0,y_0)$  and  $(x,f(x))$  of the curve, when $x$ limitlessly tends to the value $x_0$ (i.e.  $x\to x_0)$.  Due to the smoothness,

$$f(x)\to f(x_0)=y_0,$$

$$(x,f(x))\to (x_0,y_0),$$

and the slope $m$ of the secant (http://planetmath.org/SecantLine) tends to

$$\underset{x\to x_0}{lim}\frac{f(x)-f(x_0)}{x-x_0}=f^{\prime }(x_0)$$

which will be the slope of the tangent line.

Note.  Because the tangency is a local property on the curve, the tangent with the tangency point  $(x_0,y_0)$  may intersect the curve in another point, and then the tangent is a secant (http://planetmath.org/SecantLine), too.  For example, the curve  $y=x^3-3x^2$  has the line  $y=0$  as its tangent in the point  $(0,\mathrm{\hspace{0.17em}0})$  but this line the curve also in the point  $(3,\mathrm{\hspace{0.17em}0})$.

Title	tangent line
Canonical name	TangentLine
Date of creation	2013-03-22 14:50:31
Last modified on	2013-03-22 14:50:31
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	12
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 26B05
Classification	msc 26A24
Synonym	tangent
Synonym	tangent of the curve
Synonym	tangent to the curve
Related topic	Curve
Related topic	TangentOfConicSection
Related topic	Hyperbola2
Defines	tangency point