Looks like something is missing from the Limit function:
Limit[Sin[Sqrt[x^2 - 1]] - Sin[Sqrt[x^2 + 1]], x -> x0]
where x0 is set to the desired value. The function in question also has multiple values of x0 that result in a limit/value of zero (although it doesn't appear that it's necessary to use Limit - but I must be missing something).
Plot[Sin[Sqrt[x^2 - 1]] - Sin[Sqrt[x^2 + 1]], {x, -20, 20},
PlotRange -> {{-20, 20}, {-1, 0.5}}]
results in