# How do you find all the asymptotes for function f(x) = (3)/(5x)?

Jun 10, 2015

• ${\lim}_{x \to 0} \left(\frac{3}{5 x}\right) = \infty$ So there is a vertical asymptote in $x = 0$.
• ${\lim}_{x \to \infty} \left(\frac{3}{5 x}\right) = 0$ So there is a horizontal asymptote in $y = 0$.
• No oblique asymptotes.

#### Explanation:

How do we find asymptotes of f(x)?
- Vertical Asymptotes $\to {\lim}_{x \to a} \left(f \left(x\right)\right) = l$ where a is a point of discontinuity of f(x). Vertical asymptote in $x = a \Leftrightarrow l = \infty$.
- Horizontal Asymptotes $\to {\lim}_{x \to \pm \infty} \left(f \left(x\right)\right) = l$. Horizontal asymptote in $y = l \Leftrightarrow l \ne \infty$.
- Oblique Asymptotes $\Leftrightarrow$ there aren't horizontal asymptotes.

Let's verify the solutions found in this case:

graph{3/(5x) [-10, 10, -5, 5]}