# 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

 $\lim_{x\to x_{0}}\frac{f(x)\!-\!f(x_{0})}{x\!-\!x_{0}}=f^{\prime}(x_{0})$

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  $(0,\,0)$  but this line the curve also in the point  $(3,\,0)$.

 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