It is easy to think that $\displaystyle\lim_{x\to\infty}1^x$ is just $1$, but it isn't so obvious when you rewrite it as $\displaystyle\lim_{x\to\infty}(e^{2i\pi})^x$.
$e^{2i\pi x}$ is $1$ whenever $x$ is an integer, but can be a complex number if $x$ is, say, $\frac13$. Since $e^{2i\pi}=1$, $e^{2i\pi x}$ should be equivalent to $1^x$, but we usually interpret $1^\frac13$ as $1$ and $e^{\frac23i\pi}$ as a nontrivial third root of unity. This raises some questions. When taking numbers to the power of an integer, there is always a single possible value, yet when taking numbers to the power of a non-integer, there are multiple possible values. How do we precisely define $x^y$ such that $x^y$ is a single value? Is exponentiation even a function which has a single output for each input?
Back to the original limit. We interpret $1^x$ to always be equal to $1$, so $\displaystyle\lim_{x\to\infty}1^x$ should just be $1$. However, $(e^{2i\pi})^x$ isn't always equal to $1$; as $x$ changes, when we graph the expression on the complex plane, the output spins around counterclockwise in a circle. We can't find the limit when $x$ approaches infinity because the expression clearly doesn't approach anything.
Now, if we restrict $x$ such that $x\in\mathbb{Z}$, we can see that $\displaystyle\lim_{x\to\infty}(e^{2i\pi})^x$ approaches $1$ as well as $\displaystyle\lim_{x\to\infty}1^x$. How come $\displaystyle\lim_{x\to\infty}(e^{2i\pi})^x$ is defined when approaching infinity from the integers, but not when we approach it from the real numbers? How do we define infinity?
One way I came up with to define infinity is that it is the cardinality of a set $A$ such that a bijection exists between $A$ and one of $A$'s proper subsets. This definition implies that infinity is kind of like an integer that is larger than all other integers. However, under this definition, we may be able to interpret infinity to behave like an integer and conclude that $\displaystyle\lim_{x\to\infty}(e^{2i\pi})^x$ is just $1$.
What is the resolution to all of this confusion?
$(e^{2\pi i })^x$ is 1 for every $x \in \Bbb C$. The key point of your question is not about limits or infinity, but rather about the fact that exponential laws don't generally hold for complex numbers. $\left(a^b\right)^c \neq a^{bc}$ in general for complex $a,b,c$.