Question

what are the vertical and horizontal asymptotes of $f(x) = \frac{2x}{x - 1}$?
horizontal asymptote at $y = 1$, vertical asymptote at $x = 2$
horizontal asymptote at $y = 2$, vertical asymptote at $x = 1$
horizontal asymptote at $y = 0$, vertical asymptote at $x = 1$
horizontal asymptote at $y = 1$, vertical asymptote at $x = 0$

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, \( D(x)=x - 1 \), so set \( x-1=0 \), which gives \( x = 1 \). 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, \( N(x)=2x \) (degree 1), \( D(x)=x - 1 \) (degree 1)), the horizontal asymptote is the ratio of the leading coefficients. Leading coefficient of \( N(x) \) is 2, leading coefficient of \( D(x) \) is 1, so horizontal asymptote is \( y=\frac{2}{1}=2 \).

Answer:

horizontal asymptote at \( y = 2 \), vertical asymptote at \( x = 1 \) (the middle option with this description)