# How to find the asymptotes of f(x)= (3e^(x))/(2-2e^(x))?

Oct 3, 2017

Horizontal asymptotes $y = 0$ and $y = 1.5$ and vertical asymptote $x = 0$

#### Explanation:

One can find asymptotes of $f \left(x\right)$ two ways

one , finding how $f \left(x\right)$ behaves or tends as $x \to \infty$ or $x \to - \infty$

and here as $f \left(x\right) = \frac{3 {e}^{x}}{2 - 2 {e}^{x}} = \frac{3}{2 {e}^{- x} - 2}$

as $x \to - \infty$, ${e}^{x} \to 0$ and $f \left(x\right) \to \frac{0}{2} = 0$.

Similarly as $x \to \infty$, ${e}^{- x} \to 0$ and $f \left(x\right) \to \frac{3}{2} = 1.5$

This gives us horizontal asymptotes as $y = 0$ and $y = 1.5$

two as $f \left(x\right) = \infty$, $2 - 2 {e}^{x} = 0$ or ${e}^{x} = 1$ i.e. $x = 0$

Hence we have a vertical asymptote $x = 0$

graph{(3e^x)/(2-2e^x) [-10, 10, -5, 5]}