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Horizontal and Slant (Oblique) Asymptotes

Check it out:

Find the horizontal asymptote for

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

If we write out the long division...

( 4x^2 - 3x + 1 ) / ( 2x^2 - 1 ) = 2 ... this gives 4x^2 - 2 ... subtracting gives -3x yadda
Done!

We got this 2 by looking at 4x^2 / 2x^2 = 2

So, the horizontal asymptote is the line

y = 2

We could have just taken a quick look at

4x^2 / 2x^2

and have been done with it!