Finding the order of magnitude of functions as ϵ tends to zero

Question

find the order of magnitude of the function sinh $\frac{1}{ϵ}$ as ϵ tends to zero. The answer is suposed to be in terms of exponential function but when you look at the maclaurin series and replace the x with $\frac{1}{x}$ it becomes infinite and there is no answer??

Answer 1

Hint:

$$\sinh u=\frac{\mathrm e^u-\mathrm e^{-u}}2$$ and set $u=\dfrac1x$.