Oblique Asymptotes (1 Viewer)

[mantiswhoprays]

can someone give me an overview of how to work out/with oblique aymptotes, i was away in class when this was done. when i got back the teacher gave me a quick overview but it went completely over my head cause she rushed through it. thanks

 

[tommykins]

You can either do partial fractions (splitting it up) or dividing the polynomial.

ie. x² - 1 / x² - 9

= x² - 9 + 8 / x²-9 = x²-9/x²-9 + 8/x²-9 = 1 + 8/x²-9 Therefore, y = 1 would be an assmyptote.

Sorry I can't really think of an oblique one at the moment, but you should get the drift.

 

[Slidey]

You have a function of the form:
y = (ax^2+bx+c)/(dx+e), then do the following:

y = (x^2+x+1)/(2x+3)
y = (x^2+3/2x - 3/2x + x+1)/(2x+3)
y = (1/2)x + (-(1/2)x+1)/(2x+3)
y = x/2 + (-x/2-3/4 +3/4+1)/(2x+3)
y = x/2 - 1/4 + (7/4)/(2x+3)

Oblique asymptote at y=x/2-1/4

Or simpler:
y = (x^2-x+2)/(x+1)
y = (x^2+x -x -x+2)/(x+1)
y = x -(2x-2)/(x+1)
y = x -(2x+2 - 2 - 2)/(x+1)
y = x - 2 + 4/(x+1)

Oblique asymptote is: y=x-2

This is essentially polynomial division.

You get oblique linear asymptotes whenever the degree of the numerator is one more than the degree of the numerator, e.g.:
y=(x^3-1)/(x^2+1), asymptote at y=x

But these higher degree ones require explicit polynomial division.

 

[mantiswhoprays]

thank you so much, heaps clearer now. i've got my ext 1 exam tomorrow, feeling prepared now!!! thanks again

 

[foram]

oblique asymtopes are 4U math though... i thought...

 

[Trebla]

Nope they're in the 2 unit syllabus, though I expect that the asymptote would be derived formally for a sub-question rather than expecting you to find the asymptote without explicitly asking for it.

Other than the polynomial division, the easiest way to find oblique asymptotes is to find the limit as x approaches infinity and negative infinity. It is particularly useful if the function is not a polynomial. For example, y = x + e^x, has an asyptote at y = x, because as x approaches negative infinity, e^x approaches zero, however this asymptote is only valid for large negative x, and does not apply for large positive x.

 

[mantiswhoprays]

turns out i didn't even need to know this for my exam, still handy to know for next time though.

 

[advanced sam]

can someone give me an overview of how to work out/with oblique aymptotes, i was away in class when this was done. when i got back the teacher gave me a quick overview but it went completely over my head cause she rushed through it. thanks

is this for 4 unit

 

[Slidey]

As others have said: not really.

I'm pretty sure I learnt oblique asymptotes in 3unit.