I expanded everything out and got [3x^2 + x^2*sin(x)] / [x^2 + 2sin(x) + sin^2(x)] and I tried splitting the problem but it didn't work because the denominator is always 0 when I try plugging in '0' for 'x.' I can't find any way to cancel out the denominator
With this kind of limit, I like to first note which factors are relevant. That (3+sin(x)) part isn't - it's going to 3, so isn't going to be the cause of the limit converging or not. The other two factors are what count - the x-squared in the numerator, and the (x+sin(x)) squared in the demoniator, are what is making this be of the form 0/0; that's where the action is; that's where you need to pull some trick or simplification to find the limit.
As a hint, when it's "like 0/0", and has x's and sin(x)'s and is a limiti as x goes to 0, almost surely you'll end up using sin(x)/x goes to 1 as x goes to 0. You could even do the following:
"Replace" sin(x) by (sin(x)/x)x, and then keep that (sin(x)/x) together in your algebra, as it'll be going to 1. Now you don't have anymore sin(x)'s running around - only x's and that (sin(x)/x) that's going to 1.
For this problem, a big hint would be to observe that:
Before I wrote "As a hint, when it's "like 0/0", and has x's and sin(x)'s and is a limiti as x goes to 0, almost surely you'll end up using sin(x)/x goes to 1 as x goes to 0.".
When I see something like this problem, after deciding that the sin(x) going to 0 is an essential reason that the limit's value is unclear, I think to myself: "I need to get an x under that sin(x)." In this case, the way to do that was to divide both numerator and demoninator by x.