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According to definition, secant lines intersect the curve on two different points say $P,Q$ while tangent lines intersect only at one point. Also according definition with $P$ fixed and $Q$ variable as $Q$ approaches $P$ along the curve direction of secant approaches that of tangent.

Now my question is if is curve like sine curve then can we find a tangent line on arbitrary point if so i guess it will contradict the definition of tangent line that it only intersect only a single point of the curve

Please help Ahsan

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    $\begingroup$ The number of intersections is a local condition. A line is the tangent line to a curve at the point $P = (x, y)$ if there is an interval (possibly very small) around $x$ such that the line only intersects the curve once in this interval. The tangent line can intersect the curve arbitrarily often in other places. $\endgroup$ – Sammy Black Mar 17 '14 at 2:09
  • $\begingroup$ What about $f(x)=x^2\sin\left(\frac1x\right)$ (with its discontinuity filled in) at the origin? The $x$-axis is a tangent line, because it has the right slope, and the difference between the function and it shrinks faster than any linear function. However, there's no interval on which it only intersects the curve once. $\endgroup$ – G Tony Jacobs Mar 17 '14 at 2:12
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The definition of tangent is not that it just intersects at one point. It has to do with precisely the way the line touches the curve at that point, and nothing to do with what happens anywhere else.

If you zoom in closer and closer to the point of tangency, and as you get closer, the curve and the line become indistinguishable, then it's a tangent line. It doesn't matter how many times it might contact the curve at other points, as long as it matches at the point we're interested in.

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  • $\begingroup$ then what is difference between secant lines and tangent lines $\endgroup$ – Ahsan Iqbal Mar 17 '14 at 9:31
  • $\begingroup$ If you zoom in close to where the secant line meets the function, it looks like two lines crossing, or something other than two copies of the same line. $\endgroup$ – G Tony Jacobs Mar 17 '14 at 13:14
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If it were correct to define the concept of tangent line by saying it's a line that intersects a curve at only one point, then the $y$-axis would be a tangent line to the curve $y=x^2$. That is absurd.

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