Find all the asymptotes

Question

For the following functions, use "x" to indicate that the -axis is an asymptote, "h" to indicate a horizontal asymptote other than the x-axis, "v" to indicate a vertical asymptote, "s" to indicate a slanted asymptote, and "n" the lack of an asymptote. If the graph of a function has several types of asymptotes indicate them all in alphabetical order.

f(x)=x/(x-1)²

f(x)=x²/(x-1)²

f(x)=x³/(x-1)²

f(x)=x²/(x²+1)

 

Answer 1

The rules are:

 

if the degree of the denominator (polynomial on the bottom) is greater than the degree of the numerator (polynomial on top) ... you get horizontal asymptote at x-axis (y=0); and vertical asymptote when the denominator equals 0 ... in the first three problems, x=1; in the last problem,

 

if the degrees of the numerator and denominator are the same, you get horizontal asymptote by dividing the coefficient of the numerator's highest degree term by the coefficient of the denominator's highest degree term; vertical asymptotes, same as first case.

 

if the degree of the numerator is higher than the denominator BY 1, you get slant asymptote = result of dividing the numerator by the denominator AND OMITING ANY REMAINDER ... no horizontal asymptote; as before, vertical asymptote when the denominator equals 0.

 

NOW IT'S YOUR TURN ... HAVE FUN SOLVING!!