Pregunta
Find the horizontal asymptote of \( f(x)=\frac{4 x^{3}-x-3}{-5 x^{3}-4 x^{2}+1} \) \( y=\square \)
Ask by Moran Moran. in the United States
Mar 12,2025
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The horizontal asymptote is \( y = -\frac{4}{5} \).
Solución
We compare the degrees of the numerator and denominator:
1. The degree of the numerator, \(4x^3 - x - 3\), is \(3\).
2. The degree of the denominator, \(-5x^3 - 4x^2 + 1\), is also \(3\).
Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients:
\[
y = \frac{4}{-5} = -\frac{4}{5}
\]
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To find the horizontal asymptote of the function \( f(x) = \frac{4x^3 - x - 3}{-5x^3 - 4x^2 + 1} \), we look at the degrees of the polynomials in the numerator and the denominator. Both the numerator and the denominator are of degree 3. When the degrees are equal, the horizontal asymptote is found by taking the ratio of the leading coefficients. The leading coefficient of the numerator (4) divided by the leading coefficient of the denominator (-5) gives us: \[ y = \frac{4}{-5} = -\frac{4}{5} \] Therefore, the horizontal asymptote is \( y = -\frac{4}{5} \).
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