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

Question

Sadie Ackerman · Accepted Answer

Horizontal asymptote is #y=1# 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.

Xavier Alonso · Answer

undefined To find the horizontal asymptote for the function $ y = \frac{x + 1}{x - 1} $, you compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, divide the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. In this case, the degree of the numerator and denominator is the same (1), so the horizontal asymptote is the ratio of the leading coefficients, which is $ y = \frac{1}{1} = 1 $. Thus, the horizontal asymptote is $ y = 1 $.