find all the asymptotes

Question

For the following functions, indicate that the x-axis is an asymptote, a horizontal asymptote other than the axis, a vertical asymptote, a slanted asymptote, or no asymptotes at all. 

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

f(x)= x/(x-1)^2 = 0 + x/(x-1)^2

 

y = 0, or x-axis is an asymptote (horizontal)

Can't divide by zero so x = 1 is a vertical asymptote.

 

 

f(x)=x^2/(x-1)^2 = 1 + (2x-1)/(x-1)^2

 

y = 1 is an asymptote (horizontal).
Can't divide by zero so x = 1 is a vertical asymptote.

 

 

f(x)=x^3/(x-1)^2 = x + 2 + (3x-2)/(x-1)^2

 

So y = x + 2 is a slant asymptote.

Can't divide by zero so x = 1 is a vertical asymptote.

 

 

f(x) = x^2/(x^2+1) = 1 - 1/(x^2+1)

 

So y = 1 is a horizontal asymptote.
No vertical asymptotes because x^2+1 > 0 for all real numbers.