How do you identify the oblique asymptote of f(x)= (9-x^3)/(3x^2)?

1 Answer
Apr 16, 2018

color(blue)(y=-1/3x)

Explanation:

Oblique asymptotes occur if the degree of the numerator is greater than the degree of the denominator:

To find the oblique asymptote we divide the numerator by the denominator. We only have to divide until we have the equation of a line, y=mx+b:

We can divide this using polynomial division, or, since we only need to obtain the equation of the line we could do the following.

Write the numerator as:

-x^3+0x^2+0x+9

Now divide by 3x^2

(-x^3)/(3x^2)+(0x^2)/(3x^2)+(0x)/(3x^2)+9/(3x^2)

-1/3x+0+0+3/x^2

3/x^2 is just the remainder and we don't need this. We therefore have the line equation:

y=-1/3x

This is the oblique asymptote.

The graph confirms the findings:

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