Question

How do you find the vertical, horizontal or slant asymptotes for `f(x) = 1/(x+3)`?

Answer 1

vertical asymptote x = -3
horizontal asymptote y = 0

Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation, equate the denominator to zero.

solve : x + 3 = 0 → x = - 3 is the asymptote

Horizontal asymptotes occur as `lim_(x→±∞) f(x) → 0`

If the degree of the numerator is less than the degree of the denominator, as is the case here then the equation is
always y = 0

Slant asymptotes occur when the degree of the numerator is greater than the degree of the denominator. This is not the case here so there are no slant asymptotes.

Here is the graph of the function.
graph{1/(x+3) [-10, 10, -5, 5]}