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The line you get isn’t exactly the tangent line, but if the curve isn’t too bumpy, the two lines will be pretty close.

As you reduce the distance between the points, the guess gets closer to the tangent line.

When you try to calculate the slope of a tangent line, zero wrecks your approximation process.

Does the slope of the tangent line have any meaning?

However, if you suddenly cut the string, the ball will fly off along that tangent line; in the same way, a baseball pitcher’s arm travels in an arc as he throws, but as soon as he lets go, the ball flies off on the tangent.