How do you find the horizontal asymptote for #y = (x + 1)/(x - 1)#?

1 Answer
Apr 1, 2018

Answer:

Horizontal asymptote is #y=1#

Explanation:

A vertical asymptote means when as #y->+-oo#, #x# tends to some finite number. This is simpler as we know that such a limit is brought out by the denominator, here #x-1#. As #x-1->0# i.e. #x->1#, it is apparent that #y->=-oo#. Hence #x=1# is a vertical asymptote here.

On the contrary, a horizontal asymptote means when #x->+-oo#, #y# tends to some finite number. Now as when #x->=-oo#, #1/x->0#, we divide numerator and denominator by #x#.

and #y=lim_(x->oo)(x+1)/(x-1)=lim_(x->oo)(1+1/x)/(1-1/x)#

= #1#

Hence horizontal asymptote is #y=1#.
graph{(x+1)/(x-1) [-10, 10, -5, 5]}

Note: Observe that horizontal asymptote = is there only when degree of #x# in numerator and denominator is same.