If the degree is higher on top, the limit will approach an infinity (then use common sense to figure out which one). If the degree is higher on bottom, then it will approach 0. If the degrees are the same then you use the coefficient of the highest degree of x in the numerator over the coefficient of the highest degree of x in the denominator.

So, to use one of your examples.

(3x^3-x+1)/(x+3)

the degree on top is 3, and on bottom it is 1. So the limit is approaching infinity. By the looks of it, positive infinity.

Another example:

(x^2-5)/(10x^2+x+92)

degree on top: 2

degree on bottom: 2

coefficients over each other: 1/10 (<<< limit) Now, for your other example. sin oscillates and therefore can never have a limit approaching infinity.

if i knew what i was doing then i would help