Question

Find the limit, if it exists.
limₓ → −[infinity] (x-8)/(x² + 7)

Answer 1

The limit of the function (x-8)/(x^2 + 7) as x approaches negative infinity is 0. This result comes from the fact that the denominator has a higher degree than the numerator, and when divided by the highest degree term, terms with x in the denominator approach 0.

Step-by-step explanation:

To find the limit of the function (x-8)/(x² + 7) as x approaches negative infinity, we look at the degrees of the polynomials in the numerator and the denominator. Since the degree of the polynomial in the denominator (which is 2) is higher than that of the numerator (which is 1), the limit of this rational function as x approaches negative infinity will be 0. We can further substantiate this by dividing both the numerator and denominator by x², the highest degree term in the denominator, which yields :

(1/x - 8/x²) / (1 + 7/x²)

As x approaches negative infinity, all terms that have x in the denominator approach 0, and thus the limit of the entire expression goes to 0.

step by step explanation

1. Examine the degrees of the polynomials in the numerator and denominator.
2. Divide both numerator and denominator by x², the highest power of x in the denominator.
3. Observe that terms with x in the denominator approach 0 as x approaches negative infinity.
4. Conclude that the limit of the entire expression is 0.