To describe the long-term behavior of the function [tex]\( f(x) = \frac{3x^3 + 2x - 8}{x^2 + x} \)[/tex], we need to analyze its asymptotic behavior as [tex]\( x \)[/tex] approaches infinity.
### Step-by-Step Analysis:
1. Polynomial Long Division: We start by dividing the numerator by the denominator to find any slant (oblique) asymptotes.
2. Leading Terms: Identify the leading terms of the numerator and the denominator.
- Numerator leading term: [tex]\( 3x^3 \)[/tex]
- Denominator leading term: [tex]\( x^2 \)[/tex]
3. First Division:
[tex]\[ \frac{3x^3}{x^2} = 3x \][/tex]
4. Multiply and Subtract: Multiply [tex]\( 3x \)[/tex] by the entire denominator [tex]\( x^2 + x \)[/tex]:
[tex]\[ 3x(x^2 + x) = 3x^3 + 3x^2 \][/tex]
Subtract this result from the original numerator:
[tex]\[ (3x^3 + 2x - 8) - (3x^3 + 3x^2) = -3x^2 + 2x - 8 \][/tex]
5. Next Division:
[tex]\[ \frac{-3x^2}{x^2} = -3 \][/tex]
6. Multiply and Subtract: Multiply [tex]\( -3 \)[/tex] by the entire denominator [tex]\( x^2 + x \)[/tex]:
[tex]\[ -3(x^2 + x) = -3x^2 - 3x \][/tex]
Subtract this result from the remaining polynomial:
[tex]\[ (-3x^2 + 2x - 8) - (-3x^2 - 3x) = 5x - 8 \][/tex]
7. Quotient and Remainder: The result of the polynomial long division gives us the quotient [tex]\( 3x - 3 \)[/tex], and the remainder:
[tex]\[ \frac{5x - 8}{x^2 + x} \][/tex]
### Determining the Asymptote:
- As [tex]\( x \)[/tex] approaches infinity, the remainder term [tex]\( \frac{5x - 8}{x^2 + x} \)[/tex] approaches [tex]\( 0 \)[/tex].
Therefore, the function [tex]\( f(x) \)[/tex] behaves like [tex]\( 3x - 3 \)[/tex] for very large values of [tex]\( x \)[/tex], indicating a slant asymptote at [tex]\( y = 3x - 3 \)[/tex].
Given this, we analyze the options provided:
a) Horizontal asymptote at [tex]\( y=3 \)[/tex]: This is incorrect because [tex]\( f(x) \)[/tex] grows without bound.
b) Slant asymptote with a slope of 3: This is correct because the slant asymptote [tex]\( y = 3x - 3 \)[/tex] indeed has a slope of 3.
c) Slant asymptote at [tex]\( y = 3x \)[/tex]: This is incorrect because the correct asymptote is [tex]\( y = 3x - 3 \)[/tex].
d) None of these answers are correct: This is incorrect because option b is correct.
### Conclusion:
The correct answer is:
b) Slant asymptote with a slope of 3
