Why does an exponential function eventually get bigger than a quadratic

I have seen the answer to this question and this one.

My 7th 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 3x vs. 10x2 that he can just try different integer values of x until he finds one, e.g. x=6. But, how can a 7th grader understand that it will always be true, even 1.0001x will eventually by greater than 1000x2? 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.

Answer

If you have a quadratic polynomial f(x) and an exponential function b(x)=bx where b>1, you can show that b(x) surpasses the polynomial by showing that eventually the growth rate of b(x) exceeds the polynomials growth rate.

Since the polynomial’s leading term has the biggest effect when x grows very big, that means that the other terms don’t matter, so let f(x)=ax2 for an a>0. Now, compute the ratio between f(x+1) and f(x): f(x+1)f(x)=a(x+1)2ax2=(x+1x)2.
As you can see, as x grows very big, that ratio between x+1 and x grows close to 1, thus f(x+1) barely increases from f(x) because it is being multiplied by a number close to 1. Now analyze the ratio between b(x+1) and b(x). By definition, the exponential function multiplies by its base b evey time you increase by 1, so the ratio between b(x+1) and b(x) is always b. However, we stated already that b>1. We also found out that f(x) ratio approaches 1 as x gets really big. Thus, there is a point when b(x) ratio exceeds f(x) ratio, which means that b(x) will start growing faster than f(x) and will eventually outgrow f(x).

Note that I just used terminology like approach and really big because a 7th grader would not know of limits, so don’t nitpick that.

Secondary note: I said a>0 and b>1 because I assumed that both of the functions would be traveling upwards as you moved right along the x-axis.

If you want a downward-facing parabola with a<0 and a downward-facing exponential with 0<b<1, then you can just note that the exponential will just tend towards 0 when x gets very big, but the quadratic will eventually go below zero if it is facing downwards, thus showing that the exponential will eventually become greater then the quadratic.

Attribution
Source : Link , Question Author : John L , Answer Author : Jack Lilhammers

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