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 tangent¹) 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 $$\lim_{x\to x_0}\frac{f(x)\!-\!f(x_0)}{x\!-\!x_0} = f'(x_0)$$ 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 $(3,\,0)$

Footnotes

...tangent¹

The word is initially a participial form tangens (its genitive: tangentis) of the Latin verb tangere `to touch'.