How can a point have a slope? [duplicate]

You get the derivative of $f(x)$ by getting

the limit as $h$ tends to $0$ of $\dfrac{f(x+h) - f(x)}{(x+h) - (x)}$

I understand that the value of the derivative is the slope of the graph of the function at $x$. However when $h = 0$ you have just one point and you need $2$ points for a slope, don't you?

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Which is why we take the limit rather than taking $h=0$. –  Thomas Andrews Dec 3 '11 at 22:56
But in differentiating from first principles you treat any hs left over after cancelling as 0 –  jjb Dec 3 '11 at 22:59
Related to this previous answer. It's essentially the same as Berkeley's objection (if it's two points, then it's not the tangent line; if it is one point, then you aren't defining a line). See if that answer settles your doubts. –  Arturo Magidin Dec 3 '11 at 23:00
@Arturo: I ran into an old comment of me asking you why you don't have handy .pdf files prepared like Pete L. Clark; but lately it seems that at long last this website offers a suitable replacement! :-) –  Asaf Karagila Dec 3 '11 at 23:02
@jaques: To be precise, "the slope of the graph" is defined to be "the slope of the tangent line to the graph". The tangent line to the graph is defined in terms of the relative error (see the previous answer I linked to above). The derivative at a point is defined to be the slope of the tangent line to the graph at the point, or equivalently, as the limit above, because we can show that this limit, whatever it is, must be the slope of the tangent line to the graph at that point. –  Arturo Magidin Dec 3 '11 at 23:05
Note that with one point you cannot just use a formula that uses the rate of change of $x$ because you would divide by zero. That is why we take the limit of $h$ going to zero.