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Hometangent line

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# 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 tangent^{1}^{1}The word is initially a participial form tangens (its genitive: tangentis) of the Latin verb tangere ‘to touch’.) 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 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, 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 cuts the curve also in the point $(3,\,0)$.

## Mathematics Subject Classification

26B05*no label found*26A24

*no label found*

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