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

Mar 1, 2016

There is a horizontal asymptote: $y = 4$

There is a vertical asymptote: $x = 3$

Explanation:

You can rewrite the expression.

$\frac{4 x}{x - 3} = 4 \cdot \frac{x}{x - 3}$

$= 4 \cdot \frac{\left(x - 3\right) + 3}{x - 3}$

$= 4 \cdot \left(\frac{x - 3}{x - 3} + \frac{3}{x - 3}\right)$

$= 4 \cdot \left(1 + \frac{3}{x - 3}\right)$

$= 4 + \frac{12}{x - 3}$

From this, you can see that

${\lim}_{x \to \infty} \frac{4 x}{x - 3} = {\lim}_{x \to \infty} \left(4 + \frac{12}{x - 3}\right) = 4$

Similarly,

${\lim}_{x \to - \infty} \frac{4 x}{x - 3} = {\lim}_{x \to - \infty} \left(4 + \frac{12}{x - 3}\right) = 4$

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,

${\lim}_{x \to {3}^{-}} \frac{4 x}{x - 3} = - \infty$

Similarly,

${\lim}_{x \to {3}^{+}} \frac{4 x}{x - 3} = \infty$

There is a vertical asymptote: $x = 3$

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