Definition of a tangent I've been involved in a discussion on the definition of a tangent and would appreciate a bit of help.
At my high school and my college, I was taught that a definition of a tangent is 'a line that intersects given curve at two infinitesimally close points.' Aside from the possibility that tangent may elsewhere intersect the curve, to me, it was both intuitive and concise, but apparently, I'd have more chance of locating a set of hen's teeth than finding a similar definition online...
Has anybody else encountered such definition, or may have an objection to it (or an opinion on it, for that matter)? Thanks in advance.

Given a curve $y = f(x)$ in an $xy$-coordinate system a tangent to the curve at the point $(a,f(a))$ is a straight line ($y = mx + b$) with slope $m = f'(a)$.
I have never heard about the definition that you talk about. There are ways to "think" about what a tangent is. If you consider the definition of a derivative then it involves limits. And limits is where one would talk about stuff like things being "infinitesimally" close. 
Note that in math there isn't much room for opinions. Either the definition is correct or it is not. However, we often invent ways to think about certain definitions that make it intuitive for us. However, one always has to be careful not to make the picture that you have in your head the definition.
