Growth rate of two functions

Question

It's obvious that $x^2>2x+1$ for $x\ge 3$ - we just observe that for $x\ge3$, $3^2>2\cdot 3+1$ and the LHS grows much faster than the RHS. But how to determine: how faster does the LHS grow (and conclude from the result that the inequality indeed holds)?

Answer 1

Compare the derivatives: $(x^2)'=2x$ versus $(2x+1)'=2$. So $x^2$ grows at an increasing pace of $2x$ in contrast to $2x+1$ which grows at a constant pace of $2$. As $x$ grows larger $2x$ becomes (much) larger than $2$.