If you attempt to evaluate the limit at #x=0#, you should arrive at the following result:
#(-2(0)^3+0+1)/((0)^2+3(0)) = 1/0#
The form #1/0# is an undefined form. Often in limit problems it implies that the function tends towards #oo#.
However, before we make that assumption as the answer, we should investigate the two one-sided limits at 0 and ensure that they provide consistent results.
The easiest way to do that is to select a value of #x# "near" to #x=0# for evaluation. We are not so much interested in the value of the one-sided limit as we are in the sign of the result. One possible set to try would be #x = -1# and #x = 1/2#. (Be careful to not select values which cause the expression to remain undefined or evaluate to 0, such as -3 or 1!)
#f(-1) = (-2(-1)^3 + (-1) + 1)/((-1)^2+3(-1))=(2-1+1)/(1-3)#
#=2/-2=-1<0#
#f(1/2) = (-2(1/2)^3 + 1/2 + 1)/((1/2)^2+3(1/2)) = (-1/4+1/2+1)/(7/4)#
#=5/7>0#
Since our testing indicates that #lim_{x->0^-} f(x) = -oo# and #lim_{x->0^+} f(x) = +oo#, and since both one-sided limits are not equal to each other, the limit does not exist.
A graph confirms our analysis:
graph{(-2x^3+x+1)/(x^2+3x) [-10, 10, -5, 5]}
