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Does a line have a tangent line? It would be the line itself. The slope of the tangent line is the slope of the line itself. This is verified by a derivative example. But, isn't the definition of a tangent line "a line that touches at one point"? Doesn't the tangent line to a line touch at infinite points?

$f(x) = 2x+1$

$f'(x) = 2$

This implies a line does have a tangent, right?

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    $\begingroup$ No, that is not a definition of the tangent line. $\endgroup$ – PVAL-inactive Sep 16 '15 at 21:26
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    $\begingroup$ The definition does not care about other points. $\endgroup$ – Ivan Neretin Sep 16 '15 at 21:28
  • $\begingroup$ That's only the definition of the tangent line TO A CIRCLE. To a function, the tangent line is just the line going through that point that shares the slope. It can touch in plenty of other places, even everywhere. $\endgroup$ – Alan Sep 16 '15 at 21:37
  • $\begingroup$ A tangent can be a curve; why else would geometricians hold forth on, for example, two mutually tangent circles? Would not any curve to which a given line is tangent then be a tangent to that line? I admit the impossibility, though, of two straight lines being mutually tangent in the ordinary sense. $\endgroup$ – Senex Ægypti Parvi Sep 16 '15 at 21:42
  • $\begingroup$ The tangent line at a point $x$ is the limit of the line between two points on the curve, as the two points get closer to $x$. $\endgroup$ – Akiva Weinberger Sep 16 '15 at 21:52
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It's not so simple to define what is a tangent straight line to a curve. Your definition ''the line that touch at one point'' can work only if the curve is a conic section ( so it is represented by a second degree equation), but also for a simple curve as $y=x^3$ you can see that all the straight lines that are intuitively tangent at some poin, meet the curve at another point, and the only line that ''touch at one point'' is the line $y=0$ that really is a tangent, but through the curve just at the point of tangency.

And you can also easily find functions that has the same line as an ''intuitive'' tangent at more than one point (as a simple exemple: $y=\sin x$ with the lines $y=\pm1$).

So a good definition of tangent can be done only at the some time as the definition of derivative, using a limit. In words:

The tangent line is the limit position of a secant line when the two points of intersection coincide.

Developing this intuition we come to the classical definition of the slope of the tangent as: $$ m=\lim_{h\rightarrow 0}\dfrac{f(x+h)-f(x)}{h}=f'(x) $$

Now, if the curve is a stright line, this definition can also be used and, yes, the tangent to a stright line is the same stright line.

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    $\begingroup$ Ok, so tangent to a line is a line, and tangent doesn't mean "hits at one point". $\endgroup$ – JackOfAll Sep 17 '15 at 14:23
  • $\begingroup$ But this limit definition implies the 2 secant points never touch. That is not a tangent, it is still technically a secant. $\endgroup$ – JackOfAll Sep 20 '15 at 17:32
  • $\begingroup$ Technically the line passes always between two points, and is this the motif why it's always well defined, but the limit process is such that we can reduce the distance between such points as little as we want ( this is what means $h\rightarrow 0$). $\endgroup$ – Emilio Novati Sep 20 '15 at 19:24
  • $\begingroup$ But, I don't think that is a tangent line then. $\endgroup$ – JackOfAll Sep 22 '15 at 0:34
  • $\begingroup$ For a discussion about the definition of a tangent line you can read :maa.org/sites/default/files/pdf/upload_library/22/Hasse/… $\endgroup$ – Emilio Novati Sep 22 '15 at 7:51

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