Question

How do you find the limit of `(xcosx)/sinx` as x approaches 2?

Answer 1

In finding limits always try substitution first.

Explanation:

`2` is close to `(2pi)/3` so `cos 2` is close to `cos((2pi)/3) = -1/2` and `sin2 ~~ sin((2pi)/3) = sqrt3/2`

We conclude that `cos 2 != 0` and `sin 2 != 0`.

Theerfore, we do not get an indeterminate form when we substitute.

`lim_(xrarr2) (xcosx)/sinx = (2cos2)/sin2` `" "` Which may also be written `2cot2`