Examples with the hierarchy

Example — Comparison between powers and exponentials

$\lim_{x\to+\infty}\frac{x^3+2x}{e^x} = 0, \qquad \text{perché } x^3 \ll e^x.$ $\lim_{x\to+\infty}\frac{3^x}{x^{100}} = +\infty, \qquad \text{perché } x^{100}\ll 3^x.$

In the first case the denominator (exponential) grows infinitely faster than the numerator (polynomial): the ratio tends to $0$. In the second case it is the numerator (exponential) that prevails over any power, even $x^{100}$: the ratio tends to $+\infty$.

Links

Topics: Limits
Concepts: Hierarchy of infinities
Methods: Limits by comparison
Skills: Computing limits