Limit calculation , taylor or L'hopital ?

[akleron]

Hello !
I'm trying to figure out what Technic should I use with while calculating the following limits.
Honestly I have no idea what signs should I look for.
Would love some guidance!
thanks
1.

2.

 

[Dr.Peterson]

Give l'Hopital a try, and show us your work as far as you get. That's the best way we can help.

 

[akleron]

Hey
Eventually I went for Taylor with the 2nd limit because of the x->0 limit which allows me to use Maclaurin series.
According to Wolfarm I got the answer right but I got no clue if I did it properly (especially with the piano ("o") remaining )
Would love you guys to tell me what you think !

 

[lookagain]

akleron, don't box that final expression in post # 3. Evaluate \(\displaystyle \ sin(\pi/3), \ \) and simplify the complex fraction.

 

[akleron]

akleron, don't box that final expression in post # 3. Evaluate \(\displaystyle \ sin(\pi/3), \ \) and simplify the complex fraction.

Aye,
But is the calculation itself is fine ?
I'm uneasy with that piano remaining..