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

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Answer 1 · 2018-04-16T10:52:15

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

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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