What about this one?

Find the horizontal asymptote of

f ( x ) = 2x / ( x^2 - 5x - 3 )

Look at 2x / x^2 ... the denominator goes into the numerator 0 times!

So the horizontal asymptote is the line

y = 0
(which is the x-axis)

This one falls under part

 1

on our list.

 

YOUR TURN:

Find the horizontal asymptote of

f ( x ) = ( x - 3 ) / ( x^2 - 3x - 10 )


OK, so what about this one?

f ( x ) = ( 3x^3 + 2 ) / ( x^2 - x - 7 )

If we look at 3x^3 / x^2 , we find that the x^2 WILL divide in...  
But, there's going to be some x stuff left over to deal with.  This is when you need to start in with some long division...  and we get to ignore the remainder!

( 3x^3 + 0x^2 + 0x + 2 ) / ( x^2 - x - 7 ) = 3x ... this gives 3x^3 - 3x^2 - 21x ... subtract, which gives 3x^2 + yadda ... dividing again gives 3x+ 3 ... this is the slant asymptote ... it's the line y = 3x + 3

You can stop here since the rest will be remainder stuff.

graph of the slant asymptote y = 3x + 3 ... it is dashed the crosses the y-axis at y = 3 and the x-axis at x = -1


TRY IT:

Find the slant asymptote of f ( x ) = ( 5x^2 - 3x + 1 ) / ( x + 2 )