onsider the following rational functions 2x-1 x36x2 9x x 3 s(x)X+ 125 t(x) = x 2

Question

onsider the following rational functions 2x-1 x36x2 9x x 3 s(x)X+ 125 t(x) = x 2 x2-25 (a) Which of these rational functions has a horizontal asymptote? (Select all that apply.) r(x) s(x) O t(x) O u(x) (b) Which of these functions has a slant asymptote? (Select all that apply.) O r(x) s(x) O t(x) u(x) w(x) (c) Which of these functions has no vertical asymptote? (Select all that apply.) r(x) s(x) t(x) u(x) w(x) (d) Which of these functions has a "hole"? (Select all that apply.) O r(x) O s(x) O t(x) O u(x) (e) What are the asymptotes of the function r(x)? (Enter your answers as a comma-separated list of equations.) 1,2,0

Explanation / Answer

a. Those functions have horizontal asymptote whose numerator'degree is less then or equal to denomiator's degree. So the correct options are r(x) and u(x).

b. Slant asymptote

numerator's degree>denominator's degree

So the correct options are s(x),t(x),w(x)

c.No vertical asymptote

s(x) , since on setting the denominator to zero , we will not get real values.

w(x), since x+3 is common so it cancel out from the denominator.

d.hole

Only w(x) have a common term in numerator and denominator that is x+3.So correct option is w(x).

e. horizontal asymptotes, y=0

vertical asymptotes, x2-x-2=0

(x-2)(x+1)=0

x=-1,2

so the equations are

y=0 , x=-1, x=2

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