what is the limit as x goes to infinity of
(8x^3-10x^2-2x)/(3-10x-11x^3)?
what is the limit as x goes to NEGATIVE infinity of
(8x^3-10x^2-2x)/(3-10x-11x^3)?
To find the limit of a function as x approaches infinity or negative infinity, you need to determine the highest degree term in the numerator and denominator.
1. Limit as x approaches infinity:
For the limit as x goes to infinity, we analyze the highest degree terms in the numerator and denominator. The highest degree term in the numerator is 8x^3, and the highest degree term in the denominator is -11x^3. Divide both the numerator and the denominator by x^3:
(8x^3/x^3 - 10x^2/x^3 - 2x/x^3) / (3/x^3 - 10x/x^3 - 11x^3/x^3)
This simplifies to:
(8 - 10/x - 2/x^2) / (3/x^3 - 10/x^2 - 11)
As x approaches infinity, 10/x and 2/x^2 both approach zero, and 3/x^3 approaches zero faster than the other terms. Therefore, we can simplify further:
(8 - 0 - 0) / (0 - 0 - 11) = 8 / -11 = -8/11
So, the limit as x approaches infinity of the given function is -8/11.
2. Limit as x approaches negative infinity:
For the limit as x goes to negative infinity, the process is similar. Once again, we analyze the highest degree terms in the numerator and denominator. The highest degree term in the numerator is 8x^3, and the highest degree term in the denominator is -11x^3. Divide both the numerator and the denominator by x^3:
(8x^3/x^3 - 10x^2/x^3 - 2x/x^3) / (3/x^3 - 10x/x^3 - 11x^3/x^3)
This simplifies to:
(8 - 10/x + 2/x^2) / (3/x^3 - 10/x^2 - 11)
As x approaches negative infinity, both 10/x and 2/x^2 approach zero, and 3/x^3 approaches zero. Therefore, we can simplify further:
(8 - 0 + 0) / (0 - 0 - 11) = 8 / -11 = -8/11
So, the limit as x approaches negative infinity of the given function is also -8/11.