How do you find the asymptotes for #y = x/(x-6)#?

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Answer 1 · 2016-02-22T00:18:46

The asymptotes are #y=1# and #x=6#

To find the vertical asymptote, we only need to take note the value approached by x when y is made to increase positively or negatively

as y is made to approach #+oo# , the value of (x-6) approaches zero and that is when x approaches +6.

Therefore, #x=6# is a vertical asymptote.

Similarly, To find the horizontal asymptote, we only need to take note the value approached by y when x is made to increase positively or negatively

as x is made to approach #+oo# , the value of y approaches 1.

#lim_(x " "approach +-oo) y=lim_(x " "approach +-oo)(1/(1-6/x))=1#

Therefore, #y=1# is a horizontal asymptote.

kindly see the graph of

#y=x/(x-6)#.
graph{y=x/(x-6)[-20,20,-10,10]}

and the graph of the asymptotes #x=6# and #y=1# below.
graph{(y-10000000x+6*10000000)(y-1)=0[-20,20,-10,10]}
have a nice day!