The limit as x approaches negative infinity is

(3x^5 + x + 2) / (8x^4 - 5)

a. DNE
b. 0
c. -3/8
d. 3/8

I think the answer is Choice C, but I was stuck between this answer and Choice D.

lim(3x^5 + x + 2) / (8x^4 - 5) , as x ---> negative infinity

divide top and bottom by x^4
= lim (3x + 1/x^3 + 2/x^4)/(8 - 5/x^4)
when x ---> - infinity, the terms 1/x^3, 2/x^4, -5/x^4 all approach zero, so we are left with

lim 3x/8 = lim (3/8)x
now as x ---> - negative, the result becomes - infinity

I don't know what DNE stands for, but none of the other choices are correct.

DNE means Does Not Exist, so (a) is the choice of choice.

To find the limit as x approaches negative infinity, you can divide the highest power terms in the numerator and denominator by x⁴, which will help simplify the expression. By doing this, you get:

(3/x⁴ + 1/x³ + 2/x⁴) / (8 - 5/x⁴)

As x approaches negative infinity, the terms with the highest negative exponents (1/x³ and 5/x⁴) become very close to zero, because the denominator (x) becomes very large in magnitude. Consequently, we can disregard these terms because they will have minimal impact on the overall expression.

After simplifying, we are left with:

(2/x⁴) / (8 - 0)

Which can be further simplified as:

(2/x⁴) / 8

Since x is approaching negative infinity, the value of 2/x⁴ becomes arbitrarily close to zero. Therefore, the entire expression approaches zero divided by 8, which is equal to zero:

0 / 8 = 0

Hence, the correct answer is b) 0.