question find the equation of all horizontal asymptotes of the following function. f(x)=-3·(2.1)^x answer…

Explanation:

Step1: Recall exponential - function behavior

For an exponential function of the form $y = a\cdot b^{x}$, where $a\neq0$ and $b> 0,b\neq1$.

Step2: Analyze the limit as $x\to\pm\infty$

For the function $f(x)=-3\cdot(2.1)^{x}$, as $x\to+\infty$, since $b = 2.1>1$, we have $\lim_{x\to+\infty}-3\cdot(2.1)^{x}=-\infty$. As $x\to-\infty$, $\lim_{x\to-\infty}-3\cdot(2.1)^{x}=-3\cdot\lim_{x\to-\infty}(2.1)^{x}$. Let $t=-x$, then as $x\to-\infty$, $t\to+\infty$ and $\lim_{x\to-\infty}(2.1)^{x}=\lim_{t\to+\infty}(2.1)^{-t}=\lim_{t\to+\infty}\frac{1}{(2.1)^{t}} = 0$. So $\lim_{x\to-\infty}-3\cdot(2.1)^{x}=0$.

Answer:

$y = 0$