#lim_ {x->-oo} \frac{x^8-x^3+2}{-3x^4+x+7}#
Let's divide every term by the highest denominator power which is #x^4#
#lim_ {x->-oo} \frac{\frac{x^8}{x^4}-\frac{x^3}{x^4}+\frac{2}{x^4}}{\frac{-3x^4}{x^4}+\frac{x}{x^4}+\frac{7}{x^4}}#
#lim_ {x->-oo} \frac{x^4-\frac{1}{x}+\frac{2}{x^4}}{-3+\frac{1}{x^3}+\frac{7}{x^4}}#
With the algebra of limits it can be rewritten as:
#\frac{lim_ {x->-oo}x^4-\frac{1}{x}+\frac{2}{x^4}}{lim_ {x->-oo}-3+\frac{1}{x^3}+\frac{7}{x^4}}#
The first one is:
#lim_ {x->-oo}x^4-\frac{1}{x}+\frac{2}{x^4} = oo#
The second one:
#lim_ {x->-oo}-3+\frac{1}{x^3}+\frac{7}{x^4} = -3#
Hence the solution is:
#\frac{oo}{-3} = -oo#
