$\lim\limits_{x\to \infty} 1^x$

Question

Why is $\displaystyle\lim\limits_{x\to\infty} 1^x = 1$?

Why can $1^x$ be written as just $1$ when using $\lim\limits_{x\to\infty}$ and how come this does not work for $2^x$ or $3^x$ as $x\to\infty$?

Answer 1

$\forall x \in \mathbb{R}$, $1^x=1$

therefore the function $h(x)=1^x$ is constant and its value is $1$, so is the limit.

$h(x)=2^x$ is not a constant function.