Limit x -> 0 cos(x) - 1 / sin(x)

[smith007]

Homework Statement

x in the following is actually Theta

Limit as x -> 0 (cos(x) - 1) / sin(x)

Homework Equations

none

The Attempt at a Solution

Multiply both sides by cos x +1

[ (cos x - 1) / sin x ] * [ (cos x + 1 )/ (cos x + 1)]

= [ cos² x -1 ] / sin(x) (cos(x) +1)

= -sin² x / sin x (cos x + 1)

- sin x / sin x cancels to -1 leaving sin x / (cos x + 1)

Sub in x -> 0

= (-1) (sin(0) / cos(0) + 1)

= (-1) (0/2) = 0


The only step I am not sure about the is cancelation of sin x /sin x. Is this allowed? How does the solution look?

 

[Dick]

The solution looks fine. Sure you can cancel the sin(x)/sin(x).

 

[smith007]

Thank you for the fast response. Your help is much appreciated.