Limit of (x - sin x)/(x + cos x) as x tends to infinity

[quackdesk]

Homework Statement

http://prikachi.com/images/26/4273026B.gif

Homework Equations

The Attempt at a Solution

I'm having problems in solving the limit which is shown on the gif file .
As you see I use L'Hopital's rule 2 times and I get to a point when I should divide -sinx with sinx which results in -1 instead of 1 which is the right answer according to the key.

P.S sorry for posting the same thread in another sub-forum before I came here.

 

[Dick]

Homework Statement

http://prikachi.com/images/26/4273026B.gif

Homework Equations

The Attempt at a Solution

I'm having problems in solving the limit which is shown on the gif file .
As you see I use L'Hopital's rule 2 times and I get to a point when I should divide -sinx with sinx which results in -1 instead of 1 which is the right answer according to the key.

P.S sorry for posting the same thread in another sub-forum before I came here.

You should have stopped after the first differentiation. (1+cos(x))/(1-cos(x)) doesn't have indeterminant form. You can't apply l'Hopital again. And the limit of that expression as x->infinity doesn't exist. So l'Hopital doesn't apply. You'll have to think of another way to find the limit or show one doesn't exist.

 

[aeroplane]

Solving this limit may help (you can even use l'Hopital's rule!)
lim x->infinity sinx/x