Question

How do you find the asymptotes for `(4x)/(x-3)`?

Answer 1

There is a horizontal asymptote: `y = 4`

There is a vertical asymptote: `x = 3`

Explanation:

You can rewrite the expression.

`frac{4x}{x - 3} = 4 * frac{x}{x - 3}`

`= 4 * frac{(x - 3) + 3}{x - 3}`

`= 4 * (frac{x - 3}{x - 3} + frac{3}{x - 3})`

`= 4 * (1 + frac{3}{x - 3})`

`= 4 + frac{12}{x - 3}`

From this, you can see that

`lim_{x -> oo} frac{4x}{x - 3} = lim_{x -> oo} (4 + frac{12}{x - 3}) = 4`

Similarly,

`lim_{x -> -oo} frac{4x}{x - 3} = lim_{x -> -oo} (4 + frac{12}{x - 3}) = 4`

There is a horizontal asymptote: `y = 4`

You can also see that `x = 3` results in division by zero. When `x` approaches `3` from the left, the denominator will become infinisimally less than zero. So,

`lim_{x -> 3^-} frac{4x}{x - 3} = -oo`

Similarly,

`lim_{x -> 3^+} frac{4x}{x - 3} = oo`

There is a vertical asymptote: `x = 3`

Below is a graph for your reference.
graph{(4x)/(x - 3) [-40, 40, -20, 20]}