Question 115519
b)

Horizontal Asymptote:

To find the horizontal aysmptote, first we need to find the leading coefficients of the numerator and the denominator.


Looking at the numerator {{{5x}}}, the leading coefficient is {{{5}}}


Looking at the denominator {{{x-1}}}, the leading coefficient is {{{1}}}


So the horizontal aysmptote is the ratio of the leading coefficients. In other words, simply divide {{{5}}} by {{{1}}} to get {{{(5)/(1)=5}}}



So the horizontal asymptote is {{{y=5}}}





Vertical Asymptote:

To find the vertical aysmptote, just set the denominator equal to zero and solve for x


{{{x-1=0}}}



{{{x=0+1}}}Add 1 to both sides



{{{x=1}}} Combine like terms on the right side



So the vertical asymptote is {{{y=1}}}



Notice if we graph {{{y=(5x)/(x-1)}}}, we can visually verify our answers:


{{{drawing(500,500,-10,10,-10,10,
graph(500,500,-10,10,-10,10,(5x)/(x-1)),
blue(line(-15,5,15,5)),
green(line(1,-15,1,15))
)}}} Graph of {{{y=(5x)/(x-1)}}} with the horizontal asymptote {{{y=5}}} (blue) vertical asymptote {{{x=1}}} (green)