How to find the asymptotes of f(x)= 0.72^x -2?

Mar 26, 2016

$y = - 2$
I have invited another mathematician to check this

Explanation:

Given:$\text{ } f \left(x\right) = {0.72}^{x} - 2$

There are three prime conditions for this expression

Condition 1: $x > 0$

Condition 2: $x < 0$

Condition 3: $x = 0$
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$\textcolor{b l u e}{\text{Consider condition 3 }} x = 0$

When $x = 0 \text{ } f \left(x\right)$ is defined as it has a definite value.

Thus $\textcolor{b l u e}{x = 0 \text{ is NOT an asymptote}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Consider condition 2 }} x < 0$

The $f \left(x\right)$ takes the form $\frac{1}{{0.72}^{x}} - 2$

When $x = 0 \text{: "f(x)=1-2" }$ thus defined

As $x$ becomes bigger ${0.72}^{2}$ becomes smaller so

$\frac{1}{{0.72}^{2}} \text{ }$becomes bigger

Thus ${\lim}_{x \to \infty} f \left(x\right) \to \infty$ (infinity is not a number!)

$\textcolor{b r o w n}{\text{Is this an asymptote? I have invited another mathematician to look at this bit!}}$
Gut feeling is that this is NOT an asymptote!!!!!
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$\textcolor{b l u e}{\text{Consider condition 1 }} x > 0$

As $x$ becomes increasingly larger ${0.72}^{x}$ becomes increasingly smaller and smaller.

So lim_(xtooo) 0.72^x-2 -> -2color

$\textcolor{b l u e}{- 2 \text{ Is an asymptote}}$