Do it numerically.
Set your calculator to radians.
Then calculate this ratio for values of x closer and closer to 0.
x sin x / (x + sin x)
So as x approaches zero from the right, looks like the limit of the function will be 0.5
And you need to do the same from the left, so try x = -1, -0.5, -0.1, -0.01, -0.001 and see what happens to the function value. You should find the same (as x gets closer to 0 function gets closer to 0.5)
So right hand limit = left hand limit = 0.5
So the limit you need is 0.5. In calculus the limit does not need to be actually attained, you just have to be able to get as close to it as you like.