Answer
$\lim\limits_{x \to - \infty}(1.001)^{x}=0$
Work Step by Step
Evaluate limit for $\lim\limits_{x \to - \infty}(1.001)^{x}$
Consider $(1.001) =t$
Since $t\gt1$
$\lim\limits_{x \to - \infty}(1.001)^{x}=\lim\limits_{x \to -\infty}t^{x} ; t>1$
This implies
$1.001>1$
Hence, $\lim\limits_{x \to - \infty}(1.001)^{x}=0$