Question

Find the limit, if it exists, of lim x → -[infinity] (x - 9)/(x² + 5)?

Answer 1

The limit of the function (x - 9)/(x² + 5) as x approaches negative infinity is 0. This result stems from the dominance of the highest degree terms in the numerator and denominator, which lead to all other terms vanishing as x grows large in magnitude.

Step-by-step explanation:

The question is asking to find the limit of the function (x - 9)/(x² + 5) as x approaches negative infinity. To evaluate this, we can approach it by considering the leading terms in the numerator and denominator, since these terms will dominate the behavior of the function as x becomes very large in magnitude. As x approaches negative infinity, the x-term in the numerator and the x²-term in the denominator will have the largest impact on the value of the function.

Let's break it down step by step:

1. Recognize that the highest degree term in the denominator is x².
2. Divide both the numerator and the denominator by x². The function then simplifies to (1/x - 9/x²)/(1 + 5/x²).
3. As x approaches negative infinity, the terms 1/x, 9/x², and 5/x² all approach 0.
4. The resulting limit as x approaches negative infinity of (1/x - 9/x²)/(1 + 5/x²) is 0.

Therefore, the limit of the function as x approaches negative infinity is 0.