How do I find the asymptotes of a function?

Question

How do I find the asymtotes of the functions

 

y=x/x^2+2x+1

 

y=x-1/x^5-4x^3

 

y=5x^2-10x+1/x/2

Answer 1

Find the limit of the function as x approaches infinity.  To evaluate the limit, you divide the numerator and denominator by the highest power of x in the denominator.  Then evaluate when x=∞. 

 

To find the vertical asymptote, set the denominator equal to zero and solve for x.

 

 

Here is how you do the first one for horizontal asymptote.

 

lim                  x

x->∞            ____

                      x²

      ____________________   =

               x² + 2x + 1

             _____________

                    x²

 

 

      (1/x)

_________________

 1 + (2/x) + (1/x²)

 

 

Plugging in x=∞ will result in the limit being 0.   The horizontal asymptote is y=0.