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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.

As another example, if you want to find out where a ball will come to rest at the bottom of a hill, you look for a point where the tangent line is horizontal.

The steepness of the tangent line—its slope—has some important properties in physics: for instance, if you’ve got a curve that represents the position of, say, a bicycle, then the slope of the tangent line to that curve at any given point tells you how fast that bicycle is going when it reaches that spot.

Does the slope of the tangent line have any meaning?

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