What is $\lim_{x\to\infty} x^ne^{−x^2}?$ What is $$\lim_{x\to\infty} x^ne^{−x^2}?$$ for any $n$ I think its $0$ because the exponential will go faster to $0$ than the nth power go to +∞, but I don't see which theorem to apply to reach the result.
Thank you!

We have $${x^n\over e^{x^2}}=\left ( {x\over e^{x^2/n}}\right )^n$$ Therefore it suffices to show that $$\lim_{x\to \infty} {x\over e^{x^2/n}}=0$$ We can use $e^t\ge t,$  for $t\ge 0$, i.e. $e^{x^2/n}\ge x^2/n.$ Hence $$0\le {x\over e^{x^2/n}}\le {n\over x}$$
