How do you find the Vertical, Horizontal, and Oblique Asymptote given $\frac{6 e^{x}}{e^{x} - 8}$ $(6e^x)/(e^x-8)$ ?

Question

2035 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-12-01T06:20:24

The vertical asymptote is $x = ln 8$ $x=ln8$
The horizontal asymptotes are $y = 0$ $y=0$ and $y = 6$ $y=6$
No oblique asymptote

As you cannot divide by $0$ $0$

The denominator must be $\neq 0$ $!=0$

$e^{x} - 8 \neq 0$ $e^x-8!=0$

$e^{x} \neq 8$ $e^x!=8$

$x \neq ln 8$ $x!=ln8$

So the vertical asymptote is $x = ln 8$ $x=ln8$

Let $f (x) = \frac{6 e^{x}}{e^{x} - 8}$ $f(x)=(6e^x)/(e^x-8)$

$f (x) = \frac{6 e^{x}}{e^{x} - 8} = \frac{6}{1 - 8 e^{- x}}$ $f(x)=(6e^x)/(e^x-8)=6/(1-8e^(-x))$

$lim_(x->-oo)f(x)=lim_(x->-oo)6/((1-8e^(-x)))=6/-oo=0^(-)$

$lim_(x->+oo)f(x)=lim_(x->+oo)6/((1-8e^(-x)))=6$

The horizontal asymptotes are $y=0$ and $y=6$

graph{(6e^x)/(e^x-8) [-15.04, 16.99, -5.05, 10.96]}

How do you find the Vertical, Horizontal, and Oblique Asymptote given (6e^x)/(e^x-8)6exex−8(6e^x)/(e^x-8)?