Exponential beats all powers (at +∞)

The comparative growth theorem

For any integer $n \geq 1$:

$\lim_{x \to +\infty} \dfrac{e^x}{x^n} = +\infty$

Translation: at infinity, $e^x$ crushes any power of $x$.

The mnemonic

« Exp always wins over powers ». Then « powers win over logs »: $\displaystyle \lim_{x \to +\infty} \dfrac{x^n}{\ln x} = +\infty$.

Hierarchy at $+\infty$: $\ln x \ll x^n \ll e^x$.

Symmetric case at $-\infty$

$\displaystyle \lim_{x \to -\infty} x^n e^x = 0$ (for any $n$).

Exponential still crushes the power, this time toward 0.

Other key limits to remember

• $\displaystyle \lim_{x \to 0} \dfrac{e^x - 1}{x} = 1$ (derivative of $e^x$ at 0).
• $\displaystyle \lim_{x \to +\infty} \dfrac{\ln x}{x} = 0$.

Sign trap

$\displaystyle \lim_{x \to -\infty} e^x = 0$ (not $-\infty$). The exponential is always positive.

?Your turn

lim (x → +∞) (x³ / eˣ) is:

A.+∞B.0C.1D.Indeterminate

Check