# Thread: Evaluate The Following Limit. (1)

1. ## Evaluate The Following Limit. (1)

Hi

Evaluate the following limit :
$\displaystyle \lim_{x\to0} \frac{sinh(x) - sin(x)}{sin^3(x)}$
I used LHospitals, But I stucked.
I tried to use the sandwich theorem:

Clearly $\displaystyle |\frac{sinh(x) - sin(x)}{sin^3(x)}| \leq -(sinh(x)-sin(x))$
Since the minumum value for $\displaystyle sin^3(x)$ is $\displaystyle -1$.
since I make the denominator smaller then the whole fraction is bigger.
and since $\displaystyle -\lim_{x\to0} (sinh(x)-sin(x)) = 0$
then ,By using the sandwich theorem, the desired limit $\displaystyle =0$
Is this right?

Do you have another way to make the limit simpler ?

2. L'hopital's rule will work if you apply it 3 times--there might be a better way

you should get 1/3

3. Originally Posted by Calculus26
L'hopital's rule will work if you apply it 3 times--there might be a better way

you should get 1/3
Thanks.
This means my solution(using sandwich theorem) is wrong.

Still searching for professional solutions.