number theory - Proofs for $0^0 =1$?

Everyone knows the following:
$$0^x = 0 \quad \wedge \quad x^0 = 1 , \quad\forall x \in R^*$$

One morning, I wake up asking myself the question "$\text{What is $0^0$, then?}$".

So, I did what any curious highschool student would do, I tried to figure it out using algebra
$$0^0 = 0^{x-x} = \frac{0^x}{0^x} = \frac{0}{0} = \text{undefined}$$

...and I can't seem to not divide by zero every time I try.

But then, using the concept of limits, I can sort of say what it could be.

$$\lim_{x \to 0} 0^x = 0 \quad\wedge\quad \lim_{x \to 0} x^0 = 1$$

Yeah, that doesn't provide me with much clarity either.
From my perspective there's a 50% chance that it is $1$ and a 50% chance that it's $0$.
$$\text{So, probabilistically it's } ^1/_2 \text{ !??}$$

Almost all calculators that I put it into gives me back Math Error.
But oddly enough Google says otherwise

Thanks to this Numberphile video, I realize that there can't be a proper definition for it.

But many still define it to be $1$ for many more reasons. I wish to understand these reasons. I would like some proofs for this.

Hence, I seek proofs inclined (biased) to saying that $0^0 = 1$

_{EDIT: I've changed the question slightly to avoid being a duplicate.
Thank you.}

Answer

Perhaps the strongest reason why some people insist that $\;0^0=1\;$ is that

$$\text{For}\;\;0

Yet $\;x^x\;$ is undefined for lots of negative values in any neighborhood of zero...