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Calculus/sinx/x is this a continous curve

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Expert: - 2/8/2011


Question
When x = 0 is it true that sin(x)/x = 1 or is this just the case as x tends to zero. I feel the curve sin(x)/x is not continuous, but there is an infinitely small gap at x = 0.

My reason for this is that Taylor is just an approximation and not absolute.

Answer
As x approaches 0, sin(x)/x approaches 1.
It is not continuous at x = 0, but the limit does exist.

Note that sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...
Dividing this by x gives  1 - x^2/3! + x^4/3! - x^6/7! + ...

Putting x=0 into this equation gives 1, but that assumes we can divide by x,
so this assumes as close as you get to 0, the function gets closer to 1.

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