Final answer:
The function does not have a horizontal asymptote, but it approaches -1 as x approaches positive and negative infinity.
Step-by-step explanation:
The horizontal asymptote of the function (-x²-3x-1)/(x+1) can be found by examining the degrees of the numerator and denominator. In this case, the degree of the numerator is 2 and the degree of the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
However, we can still determine the behavior of the function as x approaches positive and negative infinity. To do this, we can divide the numerator and denominator by the highest power of x, which in this case is x².
Dividing both terms by x², we get (-1 - 3/x - 1/x²)/(1 + 1/x). As x approaches infinity, both -3/x and -1/x² approach 0, and 1/x approaches 0. Therefore, the function approaches -1/1, or -1, as x approaches positive infinity.
Similarly, as x approaches negative infinity, the function also approaches -1.