find the equation of all horizontal asymptotes of the following function.\n\n$f(x)=-3\\cdot(\\frac{1}{2})^x$\…

Explanation:

Step1: Recall the limit rule for exponential functions

For (y = a\cdot b^{x}), when (|b|\lt1), (\lim_{x\rightarrow+\infty}a\cdot b^{x}=0). Here (a = - 3) and (b=\frac{1}{2}) ((|\frac{1}{2}|\lt1)).

Step2: Calculate the limit as (x\rightarrow+\infty)

(\lim_{x\rightarrow+\infty}f(x)=\lim_{x\rightarrow+\infty}-3\cdot(\frac{1}{2})^{x}) Since (\lim_{x\rightarrow+\infty}(\frac{1}{2})^{x}=0) (by the property of exponential decay, as (x) gets larger, ((\frac{1}{2})^{x}) approaches (0)), then (\lim_{x\rightarrow+\infty}-3\cdot(\frac{1}{2})^{x}=0).

Step3: Calculate the limit as (x\rightarrow-\infty)

(\lim_{x\rightarrow-\infty}-3\cdot(\frac{1}{2})^{x}=\lim_{x\rightarrow-\infty}-3\cdot2^{-x}). Let (t=-x), then as (x\rightarrow-\infty), (t\rightarrow+\infty). So (\lim_{t\rightarrow+\infty}-3\cdot2^{t}=-\infty)

Answer:

One Horizontal Asymptote ((y = 0))