Question

what are the vertical and horizontal asymptotes of ( f(x)=\frac{2x}{x - 1} )?

• horizontal asymptote at ( y = 0 ), vertical asymptote at ( x = 1 )
• horizontal asymptote at ( y = 2 ), vertical asymptote at ( x = 1 )
• horizontal asymptote at ( y = 1 ), vertical asymptote at ( x = 0 )
• horizontal asymptote at ( y = 1 ), vertical asymptote at ( x = 2 )

Explanation:

Step1: Find Vertical Asymptote

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), vertical asymptotes occur where \( D(x) = 0 \) (and \( N(x)
eq0 \) at those points). Here, \( D(x)=x - 1 \). Set \( x - 1=0 \), so \( x = 1 \). Check \( N(1)=2(1)=2
eq0 \), so vertical asymptote is \( x = 1 \).

Step2: Find Horizontal Asymptote

For rational functions, if the degrees of \( N(x) \) and \( D(x) \) are equal (both degree 1 here), the horizontal asymptote is the ratio of the leading coefficients. Leading coefficient of \( N(x)=2x \) is 2, leading coefficient of \( D(x)=x - 1 \) is 1. So horizontal asymptote is \( y=\frac{2}{1}=2 \).

Answer:

horizontal asymptote at \( y = 2 \), vertical asymptote at \( x = 1 \) (the second option)