# 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 |.

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?