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It is known that there is a slant (oblique) asymptote when the degree of the numerator is 1 more than the denominator in a rational function.

To find the equation of this asymptote, you simply divide the denominator into the numerator to yield a linear function which expresses the oblique asymptote.

Consider the function $$\dfrac{x^4-16}{x^2-2x}$$

Dividing the two, you end up with the quadratic function $$x^2+2x+4$$

Graphing the two, you can see the parabola certainly behaves like an asymptote in relation to the original rational function.

So, is this an asymptote, or, something else? enter image description here

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1 Answer 1

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Rewrite the function as $$\frac{(x-2)(x+2)(x^2+4)}{x(x-2)}=\frac{(x+2)(x^2+4)}x=x^2+2x+4+\frac8x.$$ This shows the parabola $y=x^2+2x+4$ is a curvilinear asymptote to the given curve.

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