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.

By shifting and rescaling, you can turn any comparison of a quadratic vs. exponential into a standard one.  Say you are trying to determine which of $f(x) = \alpha(x-\beta)^2$ and $g(x) = ab^x$ grows faster.  By shifting $x\rightarrow x+\beta$ you can turn this into a comparison of $f_{1}(x) = \alpha x^2$ and $g_{1}(x) = a' b^x$.  Then by rescaling $x\rightarrow c x$ where $b^c = 2$, you turn the comparison into $f_2(x) = \alpha' x^2$ and $g_2(x) = a'' 2^x$.
So as long as you can convince your son that $2^x$ grows faster than $a x^2$, all other exponentials must also grow faster than all other quadratics.
