Use algebra and the properties of limits as needed to find the limit.

If the limit does not exist, say so.
(→ − ∞)lim: 9x^2+5x-17 / 5x^3+12x^2+21

The limit is zero, because

9x^2+5x-17 / 5x^3+12x^2+21 -> 9/(5x)
as x-> infinity or -infinity

To find the limit as x approaches negative infinity, we can divide the numerator and denominator by the highest power of x. In this case, it's x^3:

lim (x → -∞) [9x^2 + 5x - 17] / [5x^3 + 12x^2 + 21]

Dividing both the numerator and denominator by x^3, we get:

lim (x → -∞) [(9x^2/x^3) + (5x/x^3) - (17/x^3)] / [(5x^3/x^3) + (12x^2/x^3) + (21/x^3)]

Simplifying, we have:

lim (x → -∞) [(9/x + 5/x^2 - 17/x^3)] / [5 + 12/x + 21/x^3]

Now, we can take the limit of each term individually. As x approaches negative infinity, 1/x, 1/x^2, and 1/x^3 all approach 0:

lim (x → -∞) [9(0) + 5(0) - 17(0)] / [5 + 12(0) + 21(0)]

This simplifies to:

lim (x → -∞) 0 / 5

Since any number divided by 5 equals 0, the limit as x approaches negative infinity is:

lim (x → -∞) 0

Therefore, the limit is 0.

To find the limit of the rational function as x approaches negative infinity, we can divide the numerator and denominator by the highest power of x in the denominator, which in this case is x^3. Divide each term in the numerator and denominator by x^3:

lim(x → -∞) (9x^2 + 5x - 17) / (5x^3 + 12x^2 + 21) =
lim(x → -∞) (9x^2/x^3 + 5x/x^3 - 17/x^3) / (5x^3/x^3 + 12x^2/x^3 + 21/x^3) =
lim(x → -∞) (9/x + 5/x^2 - 17/x^3) / (5 + 12/x + 21/x^3)

Now, we take the limit of each term individually:

lim(x → -∞) 9/x = 0 (since as x approaches negative infinity, the fraction 9/x tends to 0)
lim(x → -∞) 5/x^2 = 0 (since as x approaches negative infinity, the fraction 5/x^2 tends to 0)
lim(x → -∞) 17/x^3 = 0 (since as x approaches negative infinity, the fraction 17/x^3 tends to 0)
lim(x → -∞) 12/x = 0 (since as x approaches negative infinity, the fraction 12/x tends to 0)
lim(x → -∞) 21/x^3 = 0 (since as x approaches negative infinity, the fraction 21/x^3 tends to 0)

Therefore, we have:

lim(x → -∞) (9/x + 5/x^2 - 17/x^3) / (5 + 12/x + 21/x^3) =
(0 + 0 - 0) / (5 + 0 + 0) =
0/5 =
0

Hence, the limit of the given rational function as x approaches negative infinity is 0.