$\displaystyle 3^{x+2}-1$
How do I find the asymptotes of this (vertical, horizontal)?
let f(x) = 3^x
this graph has f(x) > 0
then 3^(x+2) -1
is a transformation on f(x)
so:
f(x+2) - 1
this is a vertical translation of -1 and a horizontal translation of left 2 units
we care only about the vertical translation
now we have f(x) > -1
so the horizontal asymptote is y = -1
since the curve approaches -1 when x is large negative
Hi
Vertical asymptotes exist where the function is not defined : at the boundaries of its domain, each time that $\displaystyle \lim_{x\to x_0}f(x)=\pm\infty$, $\displaystyle x=x_0$ is a vertical asymptote for the curve.How do I find the asymptotes of this (vertical, horizontal)?
Horizontal ones exist when the function has a limit when $\displaystyle x$ approaches $\displaystyle \pm\infty$. (for example $\displaystyle y=0$ is an horizontal asymptote as $\displaystyle x$ approaches both $\displaystyle -\infty$ and $\displaystyle \infty$ for $\displaystyle x\mapsto\frac{1}{x}$)