## calculus130 3 years ago step by step using L'hopital rule: limit of 1+cos(x)/x * sin(x) as x approaches π

1. nubeer

put pi in place of x. and see what you got.

2. nubeer

hint cos pi = -1 sin pi = 0

3. calculus130

i get 0/0 if i plug pi with x

4. nubeer

yes.. now apply the derivative in numerator and seperately in denominator.

5. calculus130

i get -sinx/-xcosx

6. calculus130

after plugging the ans is 0/-1

7. nubeer

hmm please check the derivative in denominator again.. i think its wrong.. look you have to x in there so i believe you have to find derivative with product rule.

8. richyw

to compute$$\frac{d}{dx}\left(x\sin x\right)$$ you need to use the product rule.

9. richyw

$=x\cos x+\sin x$

10. richyw

$=\lim_{x\rightarrow \pi}\frac{-\cos x}{\sin x +x\cos x}$$=\frac{0}{1+\pi\cdot 0}$$=0$

11. calculus130

thnx

12. nubeer

i believe cos pi = -1 so how come got 0 in numerator.

13. richyw

wow. big mistake haha

14. nubeer

lol no worries.. :P happens sometimes with me too :P

15. richyw

or wait, sorry I meant I had the wrong numerator the numberator should be sine

16. richyw

$\lim_{x\rightarrow \pi}\frac{-\sin x}{\sin x +x\cos x}$Still works out to 0/1=0

17. richyw

or actuall $$0/-\pi$$=0 haha.