Example 2: Evaluating a Limit

If a > 1, evaluate the limit lim_x→∞ a^x.

Let H be positive infinite and a = b + 1. Then b > 0 and by inequality (vii),

a^H = (b + 1)^H ≥ bH + 1.

So a^H is positive infinite. Therefore

lim a^x = ∞.