I have to solve a trascendental equation that behaves as
-csch(ax) == x-3
which grafically looks like
As you can see, when $a$ gets bigger the csch gets closer to the axis. Since it is divergent for $x=0$, I can't plot it at $x=0$ exactly so eventually, as $a$ gets bigger and bigger I will miss the lower intersection. The red line is a plot for $a=1000000$ and, as you can see, I can't really see the lower intersection anymore, altough I know that it has to be there somewhere.
Note: The function I'm working its not a $csch(x)$ but a complicated integral that has similar behaviour. My problem is that I don't actually know that the lower intersection exists for all values of $a$. Opposite to the $csch(x)$ example where I know the function analitically, I only know the integral numerically so when $a$ gets really big I don't know if the lower intersection trully dissapears or if it is there but I need to get closer to $x=0$.
How can I improve the precision by which I obtain the lower intersection up to bigger values of a?