Lemma.
for all constant values of a.
Proof. Let 𝜀 be any positive number. Then we get:
theorem. The growth of the real exponential function x
bx exceeds all power functions,
i.e.
with a and b any constants, b > 1.
Corollary 1. lim x→0+x ln x = 0.
Corollary 2. lim x→∞
= 0.
Proof. Change in the lemma x to ln x.
Corollary 3. lim x→∞x
= 1. (Cf. limit of nth root of n.)
