consider the function f(x)=3x^2-x/x^2-6x+5

Question

List any vertical asymptotes of f(x), and explain, using limits, why each is a vertical asymptote.

List any horizontal asymptotes of f(x), and explain, using limits, why each is a vertical asymptotes.

Answer 1

Vertical asymptotes:

 

 0=  x^2 - 6x + 5 = ( x    -   5 )( x  -     1   )

 

  x-5 = 0  or x - 1 = 0

 

   vertical asymptopes at x=5 and x=1

 

 

Horizontal asymptopes:

 

   the limit as x goes to infinity results in indeterminant form infinity over infinity.

   Applying L'Hopital's rule, the derivative of the top and bottom is:

 

   6x -1  over  2x - 6

 

Again, as x approaches infinity, it is indeterminant form infinity over infinity.

 Applying L'Hopital's rule again:

 

  6/2 = 3 <--- horizontal asymptope.

 

The other way to do this to divide everything by the highest power of x which is x^2

 

  3 - 1/x  over  1 - 6/x + 5/x^2

 

as x approaches infinity, the limit is 3