Order Of Magnitude
Determine whether F(x) grows faster than x^2, same rate as x^2 or slower than x^2 as x approaches infinty.
f(x)=x^3+3

Question

Order Of Magnitude
 Determine whether F(x) grows faster than x^2, same rate as x^2 or slower than x^2 as x approaches infinty.
 f(x)=x^3+3

anonymous · Answer

If $f(x)=x^3+3$ grows faster than $g(x)=x^2$, then $f'(x)>g'(x)$, or (for $g'(x)\not=0$) $\dfrac{f'(x)}{g'(x)}>1$. (same rate: change > to =; slower rate: change > to <)

Find the limit:
$$\lim_{x\to\infty}\frac{f'(x)}{g'(x)}$$

anonymous · Answer

So I move either  x^2 to the denominator and from there I will tell what happens

anonymous · Answer

suppose $x=1,000,000=10^6$ then 
$$(10^6)^3=10^{18}$$ whereas 
$$(10^6)^2=10^{12}$$

anonymous · Answer

and 
$$\frac{10^{18}}{10^{12}}=10^6$$ so $x^3$ grows much faster than $x^2$

anonymous · Answer

I get it, so when ever I see a problem like that I can just plug in a big number and solve it.