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What is the formal definition of a tangent to a curve? The only one I can find is that it is a straight line drawn between two infinitely close points on the curve.

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same slope at point of contact –  Awesome Apr 6 at 12:07
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The evolution of mathematicians' definitions of tangency was a slow 2000 year struggle that's worth reading about. The derivative concept was what they were groping at the entire time. –  David H Apr 6 at 12:12
    
If $f$ is differentiable at $x$, then the tangent line to the graph of $f$ at $(x,f(x))$ is, technically, defined to be the line through $(x,f(x))$ whose slope is $f'(x)$. Note that it's impossible for two distinct points to be "infinitely close" to each other, so the definition you mentioned doesn't make sense. –  littleO Apr 6 at 12:16
    
@littleO, we have over 50 questions under the "infinitesimals" tag. If you think this "doesn't make sense", you could start a thread at meta to have them all deleted. –  user72694 Apr 7 at 12:53
    
@user72694 Probably I should have said that it's impossible for two distinct points on a curve in $\mathbb R^2$ to be infinitely close to each other, and noted that one can make sense of such ideas using nonstandard analysis (as you explain in your answer). I haven't studied nonstandard analysis but I'd like to learn more about it. –  littleO Apr 7 at 20:24

6 Answers 6

The line tangent to a the graph of a differentiable function at a point is the graph of the local linear approximation of the function at that point.

"Differentiable" means exactly "locally linearly approximable," so this makes sense.

It may interest you that the word "tangent" ultimately comes from the Latin tangens, present participle of tangere "to touch", so meaning "touching".

The word "secant" likewise comes from the Latin secans, present participle of secāre "to cut", so meaning "cutting".

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The OP seems to be correct in assuming that the most intuitive approach to the tangent line is through a pair of infinitely close points on the curve. To be completely precise, one needs to take the shadow of that line to obtain the tangent line; i.e., the line through a pair of infinitely close points is infinitesimally off the tangent line. To put it another way, one "rounds off" the line through a pair of infinitely close points to the nearest real slope to get the tangent line. See Keisler for details.

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Assuming the graph's function $\;f\;$ is differentiable at a point $\;(x_0,f(x_0))\;$, the tangent line to that graph at his point is defined to be the straight line

$$y-f(x_0)=f'(x_0)(x-x_0)$$

Why is it called "tangent line" and more is explained, usually with strong geometric intuition and diagrams, in basic calculus courses and books.

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There is not one formal definition; it varies from context to context which definition is most useful, and sometimes the possible definitions are not exactly equivalent for all points and all curves.

A reasonable "default" definition would be that a tangent to a curve would be that a tangent is a line that passes through a point on a curve and lies in the direction of the derivative of the curve at that point.

This definition presumes that the curve is parameterized such that it has a nonzero derivative at the point in question. There are possible ways to deal with points where the derivative is zero (such as reparameterizing by arc length, possibly considering only one-sided derivatives, and so forth), but that definitely gets us into context-specific territory.

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Generally the common definition is to define the tangent line as the limit of a secant line between two points $f(x)$ and $f(x_0) = f(x+h)$, as $h$ goes to 0 (and $x_0 \to x$).A good discussion of the tangent line can be found in this article from the MAA:

https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020721.02p01112.pdf

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A tangent line to the function $f(x)$ at the point $x=a$ is a line that just touches the graph of the function at that point and is “parallel” (in some way) to the graph at that point. Another definition, is that if the line $\rm AB$ is a secant line to the function $f(x)$ at the point $\rm A$, then the tangent line of that function at the point $\rm A$ is the limit as the point $\rm B$ approaches the point $\rm A$.

I hope this helps.
Best wishes, $\mathcal H$akim.

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