find the horizontal asymptote of $f(x)=\\frac{5x - 2x^{3}+4}{3x^{3}-5x^{2}+1}$. $y=$
find the horizontal asymptote of $f(x)=\\frac{5x - 2x^{3}+4}{3x^{3}-5x^{2}+1}$. $y=$
Answer
Explanation:
Step1: Determine the degrees of numerator and denominator
The degree of the numerator (n) (for (5x - 2x^{3}+4)) is (n = 3) (highest - power term ( - 2x^{3})). The degree of the denominator (m) (for (3x^{3}-5x^{2}+1)) is (m = 3).
Step2: Use the rule for horizontal asymptotes when (n=m)
When (n = m), the horizontal asymptote (y=\frac{a_{n}}{b_{m}}), where (a_{n}) is the leading coefficient of the numerator and (b_{m}) is the leading coefficient of the denominator. For the numerator (5x - 2x^{3}+4), the leading coefficient (a_{n}=-2). For the denominator (3x^{3}-5x^{2}+1), the leading coefficient (b_{m}=3). So (y = \frac{-2}{3}).
Answer:
(y=-\frac{2}{3})