Question

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

Answer 1

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`.