Why does an exponential function eventually get bigger than a quadratic I have seen the answer to this question and this one.
My $7$th grade son has this question on his homework:
How do you know an exponential expression will eventually be larger than any quadratic expression?
I can explain to him for any particular example such as $3^x$ vs. $10 x^2$ that he can just try different integer values of $x$ until he finds one, e.g. $x=6$. But, how can a $7$th grader understand that it will always be true, even $1.0001^x$ will eventually by greater than $1000 x^2$? They obviously do not know the Binomial Theorem, derivatives, Taylor series, L'Hopital's rule, Limits, etc,
Note: that is the way the problem is stated, it does not say that the base of the exponential expression has to be greater than $1$. Although for base between $0$ and $1$, it is still true that there exists some $x$ where the exponential is larger than the quadratic, the phrase "eventually" makes it sound like there is some $M$ where it is larger for all $x>M$. So, I don't like the way the question is written.

We can show a more general result: a (growing) exponential grows faster than any power. Eventually,
$$a^x>x^b.$$
Indeed, taking the logarithm, this is equivalent to
$$x\log a>b\log x$$
or
$$\frac x{\log x}>\frac b{\log a}.$$
The LHS is an unbounded function and the RHS a constant. (For $x>e$,
$$\frac{ex}{\log ex}=e\frac{x}{\log x+1}=e\frac{\log x}{\log x+1}\frac{x}{\log x}\ge\frac e2\frac{x}{\log x}.)$$

