Asymptotic expansion of $\exp(-ax)\,\cosh(bx)$ or $\exp(-ax)\,\sinh(bx)$

Question

I would like to understand the behaviour of

$$\exp(-ax)\,\cosh(bx)$$

or

$$\exp(-ax)\,\sinh(bx)$$

for large $x$, provided that $a,b>0$ and $a>b$ or $a<b$.

Answer 1

Expand the hyperbolic function as exponentials, i.e. write the first exoression as $$\exp(-ax)\,\cosh(bx) = \frac{1}{2}\exp((b-a)x) - \frac{1}{2}exp(-(a+b)x)$$ From this you can read off the expansion depending on signs and order of $a,b$. The second expression is handled the same way.