Ending the mathematical argument behind $0^0$. 

$0^0$ is a century problem that many mathematics students find rather disturbing.
So many people have proposed their proofs to justify their various claims behind $0^0$.
Unfortunately, $0^0$ lies at the intersection of various clashing conventions, mostly due to the fact that we overload the symbol $0$ to mean both the natural number zero and the real number zero.
In discrete Mathematics, the number $0$ is considered a natural number and therefore $0^0=1$ while in analysis the number $0$ is considered a real number $\mathbb{R}$, and therefore we say $0^0$ is certainly "undefined". since students may be confused to think that $x^y$ must tend to $1$ if $x$ and $y$ both tend to $0$, which is false.
But again, to be clear, there is no argument here over any factual matters. It’s a discussion on notational convention, nothing more. For example, if we are to consider real exponents $x\in\mathbb{R}$, what is the first derivative of $\frac{d}{dx}x^a$ at $x=0$ for $a\geq{1}$
$=a.x^{a-1}=a.0^{a-1}$ for $a=1$
$=1.0^{1-1}=0^0=1$.
Now, another problem is, some students will argue that since the natural numbers are a subset of the real numbers, $\mathbb{N}\subset\mathbb{R}$, then what differentiates the $0$ in the natural number from that of the real number.
The fact of the matter is, there is no difference between the $0$ in natural and real number. But the problem is, such expressions in the real numbers is not stable with respect to limits, for example if $x^z$ approaches $1$ as $x\rightarrow{0}$ it won't be the same case for $x^y$ as $x,y\rightarrow{0}$ so its better we leave it undefined. Or imagine the case of $\log{0}$ which is undefined when using the exponential function but in analysis it is defined as $1$ when using the power series.
These are my convictions on this subject matter which was derived from arguemtns of Alon Amit and other mathematicians on Qoura.
If you have something different or more example to help broaden the understanding of our readers, it will be welcomed.