Find all of the asymptotes

Question

For the following expression indicate if the x-axis is an asymptote, if there is a horizontal asymptote, vertical asymptote, slanted asymptote, or not asymptotes at all

f(x)= 1/x

f(x)= x³/(x²-1)

f(x)=x³/(x²+1)

Answer 1

f(x)= 1/x

 

Can't divide by zero, so x = 0 is a vertical asymptote.

As x → ±∞, f(x) → 0^±, so y = 0, the x-axis, is a horizontal asymptote.

 

 

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

 

Can't divide by zero, so x = ±1 are vertical asymptotes.
As x → ±∞, f(x) → x, so y = x is a slant asymptote.

 

 

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

 

x^2+1 > 0 for all real numbers => no vertical asymptotes.

As x → ±∞, f(x) → x, so y = x is a slant asymptote.