# 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 $y={x}^{3}-3{x}^{2}$ 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 |