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When we had not learnt Calculus, we met the concept of Tangent in Circle, which was defined as the line touching the circle at ONE point.
Then, after learning Calculus, we knew that a curve could intersect with its tangent at more than one point, and a line intersecting with a curve at only one point is not necessarily a tangent. Hence, we used Limit to define tangent, which involved TWO points and we considered one approached the other to obtain the tangent.
My question is:
The definition of a tangent to a curve should be more general than that to a circle, and hence, we can say that the definition of a tangent to a circle can be derived from the definition of a tangent to a curve.
However, limit uses TWO points, even though they are very close to each other. If they overlap with each other to become ONE point, then no line occurs. So, in theory, how can we proof that the two definitions (a general curve VS a circle) are consistent?

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2 Answers 2

Limit means approaching, not coincidence. So, if you take two points on the circle, line that goes through them is not a tangent, of course. But if you make one point closer to another, that line goes closer to the tangent. If you take it even more close, then line will be also closer. And here comes the limit into the business. Just like derivatives which are based on difference of two values. But to be true derivative one of your values "kind of" approaching another, although you never say that two values are the same, so you cannot take ratio because of denominator being "zero". That's why we define derivative at the point, not two points, and that's how we define tangent as well.

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so how can we use the "limit version" of tangent to prove that the tangent to a circle touches the circle at only one point? –  Tim Feb 13 '13 at 8:21
    
because, if you assume that tangent touches circle in two points, you always can find a point between them, which is closer than other. So that new point should be "that" point, but you can continue that over and over. So the only option left is to assume that it's actually one point in form of "two points" infinitely close to each other. –  Kaster Feb 13 '13 at 8:49
    
I still have difficulties to understand. For the tangent properties of a circle, we know that the tangent just touches a circle with ONE point, but not "in form of two points infinitely close to each other". And making use of this fact, we prove that radius is perpendicular to tangent. –  Tim Feb 13 '13 at 9:08
    
That's the whole point of infinitely close stuff. Two points are so close to each other, that they actually become one. You have same issues with finding derivatives? Because, as I said, if you want to use definition of derivatives, you need to take two values - $x$ and $x+\Delta x$, where $x+\Delta x$ is infinitely close to $x$. –  Kaster Feb 13 '13 at 9:25

Take a circle in the plane and a point on the circle. Choose coordinates so that the circle has equation $x^2+y^2=1$ and the point is $P=(0,1)$. Then near $P$ the equation of the circle is given by: $$ y = f(x) = \sqrt{1-x^2} $$ and the tangent line has slope $f'(0) = 0$, so its equation is $y=1$ which is, indeed, a line touching the circle in a single point.

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This example is excellent for getting the correct tangent. But it seems that it doesn't solve my problem (TWO close points in "limit version" of tangent becomes ONE points in tangent of Circle) directly. –  Tim Feb 13 '13 at 9:05

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