Answer
$$
\lim _{x \rightarrow \infty} \frac{\left(1+2^{x}\right) / 2^{x}}{\left(1-2^{x}\right) / 2^{x}}=-1
$$
Work Step by Step
$$
\begin{aligned}
\lim _{x \rightarrow \infty} \frac{\left(1+2^{x}\right) / 2^{x}}{\left(1-2^{x}\right) / 2^{x}}&=\lim _{x \rightarrow \infty} \frac{1 / 2^{x}+1}{1 / 2^{x}-1}\\
&=\frac{0+1}{0-1}\\
&=-1
\end{aligned}
$$
