real analysis – $epsilon,delta$-proof for the limit of $log sinh{(x^2)}-x^2$ for $x rightarrow infty$

I want to use a $epsilon,delta$-proof for the existence and value for the limit of $$log sinh{(x^2)}-x^2$$
for $x rightarrow infty$.

Now, I know the definition for such proof to be $forall epsilon > 0 exists c forall x > c: left | f(x)-L right |<epsilon$.

I am struggeling to approach this problem as I am not familiar with the method for such proofs. What if the limit as $x rightarrow infty$ is $infty$ then this definition would crumble, right?