How do you find the limit of x/sin(x)xsin(x) as x approaches 0?

1 Answer
Ken C.
May 25, 2016

Manipulate the fundamental trigonometric limit to get lim_(x->0)x/sinx=1.

Explanation:

Evaluating the limit directly produces the indeterminate form 0/0 so we'll have to use an alternate method.

We know that lim_(x->0)sinx/x=1, so we'll work from there.

Note that x/sinx=1/(sinx/x), which means lim_(x->0)x/sinx can be equivalently expressed as lim_(x->0)1/(sinx/x). Using the properties of limits, this becomes:
(lim_(x->0)1)/(lim_(x->0)sinx/x

Well, lim_(x->0)1=1 for all x, and lim_(x->0)sinx/x=1 (fundamental trigonometric limit). That means we have:
1/1

Which is simply 1.

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