I need to analyse the behaviour of the following function in parametric form as t tends to positive infinity: $$ t = m(\sinh2x - 2x)$$ $$ a = m(\cosh2x - 1)$$
The question asks to show that for large t we have that a increases linearly with time.
First thing I tried to do is to analyse what should be happening to x so that t is large. And since $\sinh x$ can be expressed in terms of exponentials: $$ t = m\cdot \left(\dfrac{e^{2x}- e^{-2x}}{2} - 2x\right)$$
We can see that $e^{2x}$ will dominate all the terms and so $x$ will need to tend to positive infinity so that we obtain large value for $t$. As $x$ tends to positive infinity: $$ a = m\cdot\left(\dfrac{e^{2x}+e^{-2x}}{2} - 1\right)$$
So as $x$ becomes larger, a will tend to $\frac{e^{2x}}{2}$ which is not linear.
There must be something that I did wrong or something I just don't see in the problem. I would also appreciate if someone could give me advice on how to sketch this function in parametric form.
Thanks!