To find the limit as x approaches infinity of the given expression:

lim x->∞ (1 + x - 3x^3) / (1 + x^2 + 3x^3)

Since we are dealing with a rational function, we can use the highest degree terms in the numerator and denominator to simplify the expression:

As x approaches infinity, the terms with the highest degree dominate the behavior of the function. In this case, the highest degree terms are -3x^3 and 3x^3 in the numerator and denominator respectively.

Therefore, the limit becomes:

lim x->∞ -3x^3 / 3x^3

Simplifying further:

lim x->∞ -1

Therefore, the limit as x approaches infinity of the given expression is -1.