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Rational Functions - More on Limits

Since we understand all this horizontal asymptote junk, we can even do these guys without the graph!

Check it out:

f( x ) = ( 3x^2 - x - 2 ) / ( x^2 + 2x - 8 )

Since the horizontal asymptote is y = 3,

as x goes toward infinity , f( x ) goes toward 3

as x goes toward -infinity , f( x ) goes toward 3
 

Draw the graph to double-check.

 

TRY IT:

For each of the following, without doing the graph, find the end-tail behavior as x goes toward -infinity and as x goes toward infinity.  Double check your answers by graphing.

f( x ) = ( x^2 - 1 ) / ( 2x^2 + x - 10 )

 

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

 

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

 

f( x ) = ( 2x^2 - x - 6 ) / ( x - 1 )