Example 4

Evaluate the limit lim_x→∞ log_ax, a > 1.

Let H be positive infinite.

Then 0 = log_a 1 < log_a H, so log_a H is positive.

If log_a H is finite, say log_a H < n, then H = log_a^H < an, which is impossible because H is infinite.

Therefore log_a H is positive infinite, so

lim_x→∞ log_a x = ∞.