How do you identify the horizontal asymptote for #2/(x+3)#?

1 Answer
Feb 23, 2018

Answer:

#y=0#

Explanation:

Without using any calculus concepts, such as the limit, here are the general rules for the horizontal asymptotes of a rational function in the form of #P(x)=(R(x))/(Q(x))# (which we have here):

  1. If the degree of the numerator is less than the degree of the denominator, #y=0# is the horizontal asymptote.

  2. If the degree of the numerator is greater than the degree of the denominator, we have no horizontal asymptote, but rather a slant asymptote.

  3. If the degree of the numerator and the degree of the denominator are equal, the horizontal asymptote is #y=a/b# where #a# is the coefficient of the term of highest degree in the numerator and #b# is the coefficient of the term of highest degree in the denominator.

We have case 1 here (the degree of the numerator is #0# and the degree of the denominator is #1#); therefore, #y=0 # is the horizontal asymptote.

With limits:

Take #lim_(x->+-∞)f(x)#:

#lim_(x->+∞)(2/(x+3))=2/(∞)=0# (Dividing #2# by an increasingly large positive number yields #0# for increasing values of #x#)

#lim_(x->+-∞)(2/(x+3))=2/(-∞)=0# (Dividing #2# by an increasingly large negative number yields #0# for increasing values of #x#)

Therefore, #y=0# is the horizontal asymptote.