Sin(0) is not only equal to 0

The problem is stated incorrectly! The value that we are finding out here isn't $f(0)$ but $\displaystyle \lim_{x\to 0} f(x)$ . And even if the problem says that "its discontinuity removed at x=0", it isn't so and you cannot just use $\displaystyle \lim_{x\to 0^+}f(x)=\lim_{x\to 0^-}f(x)=f(0)$ because $f(0)$ isn't defined! The function $f(x)$ here cannot ever be continuous at $x=0$ , you can use a graph plotter to verify that, although it's quite trivial!