Twitter proof: can't touch this (exponential)
Pt En In this twitter proof we will see that no polynomial grows faster than the exponential function. Claim: the ratio $\frac{x^p}{e^x}$ tends to $0$ as $x $ tends to infinity. Twitter proof: recall that by the Taylor expansion of $e^x $ we have $$e^x = \sum_{i=0}^\infty \frac{x^i}{i!}$$ and for $x > 0$ we have $$\frac{x^p}{e^x} \leq \frac{x^p}{\sum_{i=0}^{p+1} a_ix^i} \to 0 $$ where $a_{p+1} \neq 0$ thus proving our claim. The way I like to look at this is "if the exponential has a bit of every polynomial inside, then it will grow faster than any fixed polynomial $p(x)$" (because, in particular, the exponential "has a bit" of all other polynomials that have degree higher than that of $p(x) $. Neste post vamos ver que a função exponencial cresce mais depressa que qualquer função polinomial. Proposição: o rácio $\frac{x^p}{e^x}$ tende para $0$ quando $x $ tende para infinito. Prova num tweet: sabemos pela expansão de Taylor de $e...