Just as in your question, let us define
$$
f(x) = x^{10} + x^{3} + 10x^{2}, \hspace{20pt} g(x) = x^{10}.
$$
We will explore the behaviour of these two graphs. I will leave it to you to graph them (Desmos is a great site for graphing functions) and to play around with different functions to see how their limiting behaviour changes.
Firstly, you write the following:
With a huge number, wouldn't the difference in distance between the 2 graphs also be bigger than ever before?
This is indeed true! For $f$ and $g$ above, the difference between them is
$$
f(x) - g(x) = x^{3} + 10x^{2},
$$
and this does increase as $x$ increases. Thus it is true that $f$ and $g$ do not get closer and closer as $x$ gets larger --- but this is not what limiting behaviour describes. Instead of the additive difference between $f$ and $g$, limiting behaviour is more concerned with the multiplicative difference of $f$ and $g$, namely the quotient $f(x)/g(x)$. (For Mathemticians, the formal way to describe limiting behaviour is to use Big O notation, but this may be slightly out of your reach at the moment.)
Let us plot some values to see this explicitly (values are to $4$ decimal places).
$$
\begin{array}{cccc}
x & f(x) & g(x) & f(x)/g(x) \\ \hline
1 & 12 & 1 & 12 \\
2 & 1072 & 1024 & 1.0469 \\
3 & 59166& 59049& 1.0020 \\
4 &1048800&1048576&1.0002 \\
\end{array}
$$
Just by comparing $f(x)$ and $g(x)$ for the larger values, they look "more or less the same", despite the actual difference of $f(x)$ and $g(x)$ increasing.
Let us try a more extreme example. Define
$$
f(x) = x^{10} + 999x^{9}, \hspace{20pt} g(x) = x^{10}.
$$
Let us plot some values to see this explicitly.
$$
\begin{array}{cccc}
x & f(x) & g(x) & f(x)/g(x) \\ \hline
1 & 1000 & 1 & 1000 \\
2 & 512512 & 1024 & 500.5 \\
3 & 19722366 & 59049 & 334 \\
4 & 262930432 & 1048576 & 250.75 \\
5 & 1960937500 & 9765625 & 200.8 \\
10& 1009000000000 & 10000000000 & 100.9 \\
100&1099000000000000000000&100000000000000000000&10.99\\
\end{array}
$$
Again, the values seem to get more and more similar as $x$ increases (I suggest doing this yourself with some very large numbers to see how similar they get), and the multiplicative difference between the two is decreasing.
In fact, in both cases the multiplicative difference $f(x)/g(x)$ is getting closer and closer to $1$, which must mean that $f(x)$ and $g(x)$ are getting 'more similar' in some sense. You can see this algebraically: for $x$ sufficiently large, we have
$$
\frac{f(x)}{g(x)} \approx 1 \implies f(x) \approx g(x).
$$
In other words, the difference between the functions becomes negligible, since the numbers themselves are getting bigger far quicker than the difference between them is.
If you want to evaluate these large polynomials yourself, I suggest using Python (this is my personal preference since it can deal with very large numbers easily), and the corresponding code that I used to to define the functions is:
def f(x):
return x**10 + 999*x**9
def g(x):
return x**10
You then evaluate each polynomial by writing f(1), for example. Note that the double-asterisk ** is the Python code for exponents, so use ** in place of ^.
