How do you find the vertical, horizontal or slant asymptotes for # y = 6/x#?

1 Answer
Mar 26, 2016

Answer:

we have a vertical asymptote at #x=0#
we have a horizontal asymptote at #y=0#
graph{6/x [-13.38, 16.53, -7.87, 7.09]}

Explanation:

Given: #y=6/x#
Required vertical, horizontal or slanted asymptotes?
Solution Strategy: Definition and principles governing asymptotes.

Asymptotes Rule:
Let f be the (reduced) rational function
#f(x) = (a_nx^n + · · · + a_1x + a_0)/(b_mx^m + · · · + b_1x + b_0)#

  1. The graph of #y = f(x)# will have vertical asymptotes at those values of #x# for which the denominator is equal to zero.

  2. The graph of #y = f(x)# will have horizontal asymptote if:
    a. #m > n# (the degree denominator #gt# numerator) then
    #y = f(x)# will have a horizontal asymptote at y = 0 (x-axis)
    b. If #m = n# (degree of numerator and denominator are the same),
    then #y = f(x)# will have a horizontal asymptote at #y =a_n/b_m#
    c. If #m < n# (numerator degree is larger than denominator), then the graph of y = f(x) will have no horizontal asymptote

From 1) we have a vertical asymptote at #x=0#
From 2a #m=1>n=0# thus we have a horizontal asymptote at #y=0#