You first encountered the notion of a tangent in geometry. I found the following definition of a tangent in a geometry textbook, which is probably the definition you've been carrying around in your head ever since. [br][quote]A [b]tangent [/b]to the circle [math]C[/math] is a line that contains exactly one point of [math]C[/math].[/quote][br]It is natural to try to generalize this definition to suit our purposes in calculus.
A [b]tangent [/b]to the function [math]f[/math] is a line that contains exactly one point of [math]f[/math].[br][br]This is no good. Below, plot:[br][list][*]a function,[/*][*]a point on the function, and [/*][*]the line tangent to the function at that point [/*][/list]such that the tangent contains more than one point of the function.
Calculus was famously independently invented by two mathematicians, Isaac Newton and Gottfried Leibniz, in the 1600s. Leibniz described a tangent line as a line passing between two "infinitely close" points on a curve. But what does that mean?
What mathematical tool do we use to describe two things as "infinitely close"? How does Leibniz's description relate to the definition of the derivative?
A limit. Given two points on [math]f[/math], [math](x,f(x))[/math] and [math](a,f(a))[/math], we can make those two points "infinitely close" by taking a limit as [math]x[/math] approaches [math]a[/math]. This is the idea that motivates defining [math]f'(a)[/math] (the slope of the line tangent to [math]f[/math] at [math]x=a[/math]) as [math]\lim_{x\to a}\frac{f\left(x\right)-f\left(a\right)}{x-a}[/math].
The 1828 edition of Webster's Dictionary defines a tangent as a "line which touches a curve, but which when produced, does not cut it".[br][br]This too is no good. If we were to accept this definition, functions would not have tangent lines at their inflection points. Below, plot:[br][list][*]a function,[/*][*]an inflection point of the function, and [/*][*]the line tangent to the function at that inflection point,[/*][/list]and notice that the tangent line "cuts" the function.