009A Sample Final A, Problem 6

From Math Wiki
Revision as of 19:35, 27 March 2015 by MathAdmin (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

6. Find the vertical and horizontal asymptotes of the function  

Vertical asymptotes occur whenever the denominator of a rational function goes to zero, and  it doesn't cancel from the numerator.
On the other hand, horizontal asymptotes represent the limit as goes to either positive or negative infinity.


Vertical Asymptotes:  
Setting the denominator to zero, we have
which has a root at This is our vertical asymptote.
Horizontal Asymptotes:  
More work is required here. Since we need to find the limits at , we can multiply our by


This expression is equal to for positive values of , and is equal to for negative values of . Since multiplying by an expression equal to doesn't change the limit, we will add a negative sign to our fraction when considering the limit as goes to . Thus,


Thus, we have a horizontal asymptote at on the left (as goes to ), and a horizontal asymptote at on the right (as goes to ).

Return to Sample Exam