SOLUTION: Does the function f(x)= (x^2+3)/(x^2-1) have a vertical or horizontal asymptote? If so find the equation of the asymptote(s). If not, explicitly state that there are none.

Algebra ->  Graphs -> SOLUTION: Does the function f(x)= (x^2+3)/(x^2-1) have a vertical or horizontal asymptote? If so find the equation of the asymptote(s). If not, explicitly state that there are none.      Log On


   

Question 880407: Does the function f(x)= (x^2+3)/(x^2-1) have a vertical or horizontal asymptote? If so find the equation of the asymptote(s). If not, explicitly state that there are none.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=%28x%5E2%2B3%29%2F%28x%5E2-1%29 can be transformed into equivalent expressions that will give us the answer.

VERTICAL ASYMPTOTES:
We need to look at the denominator.
f%28x%29=%28x%5E2%2B3%29%2F%28x%5E2-1%29=%28x%5E2%2B3%29%2F%28%28x%2B1%29%28x-1%29%29 tells us that
when
the function is undefined, and as x approaches those values f%28x%29 increases without bounds.
We can say f%28x%29 approaches infinity if your instructor allows that wording,
or lim%28x-%3E-1%2Cf%28x%29%29=infinity and lim%28x-%3E1%2Cf%28x%29%29=infinity if those are the symbols used in your class.
In any case highlight%28system%28x=-1%2C%22and%22%2Cx=1%29%29 are the vertical asymptotes.

HORIZONTAL ASYMPTOTES:
We need to divide. (Or just look at polynomial degrees and leading coefficients).
tells us that
as abs%28x%29 and x%5E2 increase without bounds, f%28x%29 approaches 1 ,
because the term 4%2F%28x%5E2-1%29 approaches zero.
So highlight%28y=1%29 is the horizontal asymptote.
(You could tell that the rational function f%28x%29=%28x%5E2%2B3%29%2F%28x%5E2-1%29 would have a horizontal asymptote, because the degree of numerator x%5E2%2B3 is not greater than the degree of denominator x%5E2-1 . You could tell that the asymptote would be y=1 because the degrees of numerator and denominator are the same and the ratio of the leading coefficients of numerator and denominator is 1%2F1=highlight%281%29 ).
The graph of red%28f%28x%29%29 and its asymptotes is