# Why does this rational function have a false slant/oblique asymptote?

Let's examine the following rational function: $$f(x) = \frac{3x^3+2}{x^2-x-7}$$.

Considering that the degree of the polynomial in the numerator is 1 greater than that of the denominator, it can be assumed that the function possesses no horizontal asymptote, but possess a slant, or oblique, asymptote.

As a result of long division, the slant asymptote appears to be $$y = 3x + 3$$. However, on a graph, the function $$y = 3x + 3$$ intersects with the original function, $$f(x) = \frac{3x^3+2}{x^2-x-7}$$, at the coordinate $$(-.9583333..., .125)$$.

Why is this the case? What condition prevents $$y = 3x + 3$$ from being a true asymptote of the function $$f(x) = \frac{3x^3+2}{x^2-x-7}$$ if long division produces a result declaring otherwise?

• What's the problem with the fact that the asymptote intersects the graph of the function? – mfl Feb 15 at 14:28
• @mfl How is this allowed? Is an asymptote not a line that never intersects the function itself, since the function never approaches its asymptote? What constitutes at which point(s) intersection is allowed? – Aaron Feb 15 at 14:31
• Plot the function $\frac{\sin x}{x}.$ The line $y=0$ is an horizontal asymptote and it intersects the graph at infinitely many points. One function never intersects a vertical asymptote. But we can't say anything about horizontal and oblique asymptotes. An example with oblique: $x+\frac{\sin x}{x}.$ The line $y=x$ is an oblique asymptote. And again there are infinitely many points of intersection. – mfl Feb 15 at 14:37

Notice that intersection of the function with the asymptote does not prevent it from being an asymptote. Broadly speaking, "asymptote" for $$x$$ to infinity, for example, means that the farest we go with $$x$$, the closest the function goes to the line, but it may intersect it an infinite number of times.
For example, consider $$\frac{x^2+\sin(x)}{x}$$ and look what happens. As you can see from the plot above, there are actually two vertical asymptotes at the roots of the denominator (approximately $$x = 3.19, -2.19$$) and the portion of the graph between those crosses every non-vertical line. Outside the roots of the denominator, the graph really does approach the asymptote as expected.