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I've been involved in a discussion on definition of a tangent and would appreciate a bit of help.

At my high school and at 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.

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3 Answers 3

Strictly speaking, in standard analysis, two "infinitesimally close" points are the same point. The choice of words come from the fact that even though infinitesimals are in fact zero, if you ignore that, you can still somehow use them in calculations.

So for instance the "fraction" $\frac{\text{d}y}{\text{d}x}$ is really $\frac{0}{0}$ and thus undefined. However, if you make calculations as though it were a real fraction, you will more often than not come to the correct answer. They do this in applied mathematics all the time. So thinking about infinitesimals as "really small numbers" makes intuitive sense, and gets you what you need, even though that's not how it really is.

So, the line between two infinitesimally close points on a curve is to be interpreted as the limit of secants as the two points close in on each other. If you apply this to the graph of a function $f$, then you get the definition that a tangent is a line with height and slope matching that of $f$ at a specific point.

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My interpretation is that if one was to define tangent as intersecting the curve in one point, what would the other part of a definition be (to distinguish it from the normal, and countless other lines passing through that point)? –  Tomislav Petričević Dec 6 '12 at 15:12
    
It has to intersect and be "locally paralell" to the curve at a point. A line is usually uniquely determined by two pieces of information. The real way to determine a single line is to give two distinct points. But you can just as well do it with one point and a direction. "Locally paralell" means having the same derivative. –  Arthur Dec 6 '12 at 18:18
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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.

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Here's another take on it. This looks like a good intuitive definition. You can try to formalize it as follows, for function that are nice enough. Consider a point $a\in\mathbf R$ and a map $f:\mathbf R\to\mathbf R$. If $f$ is nice enough, you can write a local approximation around $a$: $$f(a+h)=f(a)+hf'(a)+h^ng(a+h),$$ for some $n\geq2$ and some continuous map $g$ with $g(a)\neq0$. Then the tangent at $a$ to the graph of $f$ is the line with equation $y=f(a)+xf'(a)$. Since $g$ is continuous, there is a small neighborhood of $a$ on which $g$ doesn't cancel, so on this neighborhood, the only point where the graph and the tangent meet is $x$.

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