What is the equation of the slant asymptote of the rational function?
$f(x)=10 x^3-15 x^2+x-1$
A slant asymptote (also known as an oblique asymptote) of a function \(f(x)\) is a line \(y = mx + k\) such that the function approaches this line as \(x\) tends towards \(\infty\) or \(-\infty\). Mathematically, this means \(\lim_{x \to \pm \infty} [f(x) - (mx + k)] = 0\). For a rational function \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials, a slant asymptote exists if and only if the degree of the numerator \(P(x)\) is exactly one greater than the degree of the denominator \(Q(x)\). If the degree of the numerator is equal to or less than the degree of the denominator, the function may have horizontal or vertical asymptotes but not a slant one. If the degree of the numerator is more than one greater than the degree of the denominator, the function exhibits higher-degree polynomial growth and does not approach a linear slant asymptote.
The given function is \(f(x) = 10x^3 - 15x^2 + x - 1\). We can rewrite this as a rational function:
\[f(x) = \frac{10x^3 - 15x^2 + x - 1}{1}\]
Here, the numerator is a polynomial \(P(x) = 10x^3 - 15x^2 + x - 1\) of degree 3, and the denominator is a constant polynomial \(Q(x) = 1\) of degree 0. Comparing these degrees, we see that the difference between the degree of the numerator and the degree of the denominator is \(3 - 0 = 3\).
According to the condition for a slant asymptote, the degree of the numerator must be exactly one greater than the degree of the denominator. In our case, the difference in degrees is 3, not 1. As \(x \to \pm \infty\), the term \(10x^3\) dominates the function, causing it to grow at a cubic rate. A linear slant asymptote \(y = mx + k\) cannot account for this higher-order growth, as the difference between the cubic function and a linear function will also approach \(\pm \infty\) rather than zero. Therefore, we conclude that the function \(f(x) = 10x^3 - 15x^2 + x - 1\) does not have a slant asymptote.
Understanding asymptotes is essential in engineering and physical sciences for predicting long-term behavior of systems. While simple linear models often suffice, complex systems—like the trajectory of a spacecraft or the stabilization of a chemical reaction—often follow higher-degree polynomials. Knowing whether a function approaches a straight line (a slant asymptote) or grows unbounded without one helps engineers determine if their model will eventually settle into a predictable linear trend or continue to accelerate in value, guiding decisions on system design and safety limits.