limit of a hyperbolic function

Question

How to evaluate this limit without using hopital rule: $$\lim_{c\rightarrow + \infty}{\frac{\text{sinh}\sqrt{c}}{2\sqrt{x}}}$$

Here is what I have done so far:

• we know that $\text{sinh}(x)= \frac{e^x-e^{-x}}{2}.$

So applying this to the limit we find : $$L= \lim_{c\rightarrow +\infty}{\frac{e^{\sqrt{c}}}{2\sqrt{c}}}.$$ (the other part tends to zero.) So first is my approach right+would this lead me into anything?) And second how should I complete(any hints?) Thanks in advance.

Answer 1

Since the e-function increases faster than any power function, the limit is $\infty$.