I'm interested in computing pointwise values of the function $u(x) = \sinh(k-kx)/\sinh(k)$ for $x \in (0,1)$, where $k = 10^{4}$. A direct computation of course results in overflow issues due to the $\exp(k)$ factor. However, $u(x)$ only takes on values between 0 and 1, so I'm wondering if there is a clever way that this quantity can be computed. We can also write $u$ as $$ u(x) = \frac{e^{k-kx}}{e^{k}-e^{-k}} - \frac{e^{kx-k}}{e^{k}-e^{-k}}. $$
I thought to try taking logarithms first and seeing if anything simplifies, but I wasn't able to make progress. Does anyone have suggestions for how I can get numerical pointwise values of this function? I'm aware that the function is essentially 1 at $x=0$ and 0 elsewhere, but I'm curious about the original problem.