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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?

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    $\begingroup$ What's the problem with the fact that the asymptote intersects the graph of the function? $\endgroup$ – mfl Feb 15 at 14:28
  • $\begingroup$ @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? $\endgroup$ – Aaron Feb 15 at 14:31
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    $\begingroup$ 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. $\endgroup$ – mfl Feb 15 at 14:37
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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.

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  • $\begingroup$ I apparently misdefined “asymptote,” not realizing that the asymptote exclusively regards behavior approaching infinity. Thank you! $\endgroup$ – Aaron Feb 15 at 14:38
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enter image description here

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.

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