Question

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

Answer 1

`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.