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

Jan 24, 2016

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 ($+ \infty$ or $- \infty$) 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 $+ \infty$.
As $x$ approaches $0$ from the right, the function is always negative (since $x > 0$ and $x - 5 < 0$) and, therefore, it tends to $- \infty$.

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 $- \infty$.
As $x$ approaches $5$ from the right, the function is always positive (since $x > 0$ and $x - 5 > 0$) and, therefore, it tends to $+ \infty$.

Here is the graph of this function:

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