Problem
The third order Taylor polynomial $x+\frac{x^3}{3!}$ is used to approximate $\sinh x$ for $|x|\leq 2$. What is the Lagrange error bound?
The answer is $|R_3(2)|\leq2.41791$. Shouldn't the answer be $|R_3(x)|\leq2.41791$ for $|x|\leq 2$. Why are they focusing on $x=2$ when we care about the error bound over an interval?
If you plot the two functions, you will see that the error increases as you move away from $0$ (which can actually be proved analytically), therefore the error bound is given by the gap at the largest value of $x$ in the considered interval.
That's where this given solution comes from. But you are also right in your interpretation.