Question

find the equation of all horizontal asymptotes of the following function.

$f(x)=-3\cdot(\frac{1}{2})^x$

answer attempt 1 out of 2

no horizontal asymptotes
no horizontal asymptotes
one horizontal asymptote
two horizontal asymptotes

Explanation:

Step1: Recall the limit rule for exponential functions

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

Step2: Calculate the limit as \(x

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

Step3: Calculate the limit as \(x

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

Answer:

One Horizontal Asymptote (\(y = 0\))