Question

How do you find vertical, horizontal and oblique asymptotes for `f(x)= (x+5)/(x+3)`?

Answer 1

vertical asymptote x = - 3
horizontal asymptote y = 1

Explanation:

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

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

Horizontal asymptotes occur as `lim_(xtooo) f(x) to 0`

divide terms on numerator/denominator by x

`rArr (x/x + 5/x)/(x/x + 3/x) = (1 + 5/x)/(1 + 3/x)`

As x`tooo , 5/x" and " 3/x to 0`

`rArr y = 1/1 = 1 " is the asymptote "`

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