Can you tell why the growth rate of $x^2$ is that ratio?

Question

The hypothesis: $x \geq 4$ then $x^2 \leq 2^x$.

The proof: As $x$ grows larger than $4$, the right-side $2^x$ doubles each time $x$ increases by $1$ .

However, the left-side $x^2$ grows by the ratio $\left(\dfrac {x+1}{x}\right)^2$.

If $x \geq 4$ then $\dfrac {x+1}x$ cannot be greater than 1.25 . therefore $\left(\dfrac {x+1}{x}\right)^2$ can not be bigger than 1.56 .

The Question I have is: Why is the growth rate of $x^2$ the ratio $\left(\dfrac {x+1}{x}\right)^2$ ?

Um... $1.25^2$ is actually bigger than $1.56$... – user21820 Nov 04 '17 at 16:48 — user21820, Nov 04 '17 at 16:48

score 3 · Accepted Answer · answered Oct 13 '17 at 16:47

3

When $x$ increases by 1 we go from $x^2$ to $(x+1)^2$. The growth rate here is just the fraction of these two expressions:

$$\frac{(x+1)^2}{x^2} = \left(\frac{x+1}{x}\right)^2$$

answered Oct 13 '17 at 16:47

orlp

1 Answers1