pt For the following functions usex to indicate that the rax

pt) For the following functions use·x\" to indicate that the r-axis is an asymptote, to indicate a horizontal asymptote other than the r 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) f(x) f (x) f(x)- (r-1) (z-1)2

Solution

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

the degree on the denominator (namely, 2) was bigger than the degree on the numerator (namely, 1), and the horizontal asymptote was y = 0 (the x-axis).

So, Horizontal asymtote : x axis---- x

Vertcial asymtote : x =1 ------ v

No slant asymtote

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

So, Horizontal asymtote : y = x^2/x^2 =1

y =1------ h

Vertcial asymtote : x =1 ------ v

No slant asymtote

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

Vertcial asymtote : x =1 ------ v

Slant Asymtote: if the degree of the numerator is exactly one more than the degree of the denominator (so that the polynomial fraction is \"improper\"), then the graph of the rational function will be, roughly, a slanty straight line

Divide : x^3/(x-1)^2 = x+2

y = x+2 ---------- s

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

Horizontal asymtote:

y = x^2/x^2 = 1

y =1 ----h

Vertical asymtote:

none

Slant asymtote:

none