Question

How do you find the asymptotes for `g(x)= (x+3)/( x(x-5))`?

Answer 1

A function represented by a fraction of two other functions has asymptote at those points where denominator equals to `0`, while a numerator is not equal to `0`.

Explanation:

In this particular case points `x=0` and `x=5` are exactly where the denominator is `0`, while a numerator is not.

Therefore, the function is not only undefined at these two points, but goes to infinity (`+oo` or `-oo`) as its argument approaches these values. In other words, `x=0` and `x=5` are asymptotes and function behavior around these two points is asymptotic.

At `x=0` the numerator equals to `3`.
As `x` approaches `0` from the left, the function is always positive (since `x<0` and `x-5<0`) and, therefore, it tends to `+oo`.
As `x` approaches `0` from the right, the function is always negative (since `x>0` and `x-5<0`) and, therefore, it tends to `-oo`.

At `x=5` the numerator equals to `8`.
As `x` approaches `5` from the left, the function is always negative (since `x>0` and `x-5<0`) and, therefore, it tends to `-oo`.
As `x` approaches `5` from the right, the function is always positive (since `x>0` and `x-5>0`) and, therefore, it tends to `+oo`.

Here is the graph of this function:

graph{(x+3)/(x(x-5)) [-5, 8, -5, 5]}