4. Applications of Derivatives

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

66. lim (x → 0⁺) x (ln x)²

21. a. Show that ln(x) grows slower as x→∞ than x^(1/n) for any positive integer n, even x^(1/1,000,000).

22. The function ln x grows slower than any polynomial Show that ln(x) grows slower as x→∞ than any nonconstant polynomial.

110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.

c. f(x) = 10x^3 + 2x^2, g(x) = e^x

3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

e. x ln(x)

3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

g. x^3 e^(-x)

4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

g. (1.1)^x

Find the limits in Exercises 1–6.

3. lim(x→0⁺) (cox(√x))^(1/x)

23. What roles do the functions e^x and ln(x) play in growth comparisons?