Question

How do you find the horizontal asymptote for `y=(13x)/(x+34)`?

Answer 1

Find `lim_(x->+-oo) (13x)/(x+34) = 13` hence asymptote `y=13`

Explanation:

One way is to divide both numerator and denominator by `x` and look at the limit as `x->+-oo`

`f(x) = (13x)/(x+34) = 13/(1+34/x)`

So `lim_(x->oo) f(x) = 13/(1+0) = 13`

and `lim_(x->-oo) f(x) = 13/(1+0) = 13`

So the horizontal asymptote is `y = 13`

In general if `f(x) = (p(x))/(q(x))` for some polynomials `p(x)` and `q(x)` of the same degree, then `f(x)` will have a horizontal asymptote equal to the quotient of the leading coefficients.

For example:

`lim_(x->oo) (3x^2+4x-5)/(2x^2-11x+1)=lim_(x->oo) (3+4/x-5/(x^2))/(2-11/x+1/(x^2)) = 3/2`