Question

Let f(x)=(1+x−2x²−x³)/(x²+1)​

(a) Analyze lim(ₓ→[infinity]) ​f(x) and lim( ₓ→−[infinity]) ​f(x).

Answer 1

The limits of the function f(x) as x approaches positive and negative infinity are both -x.

Step-by-step explanation:

To analyze the limits of the function f(x) = (1+x−2x²−x³)/(x²+1) as x approaches infinity and negative infinity, we can examine the behavior of the function at large positive and negative values of x.

As x approaches positive infinity, both the numerator and denominator of the function become dominated by the term with the highest degree of x. In this case, the term with the highest degree is -x³ in the numerator and x² in the denominator. Therefore, the limit of f(x) as x approaches positive infinity is -x³/x², which simplifies to -x.

Similarly, as x approaches negative infinity, the highest degree terms in the numerator and denominator dominate. In this case, the limit of f(x) as x approaches negative infinity is -x³/x², again simplifying to -x.