1. Limits and Continuity

10. True, or false? As x→∞,

c. 1/x - 1/x² = o(1/x)

111. True, or false? Give reasons for your answers.

c. x = o(x + ln(x))

111. True, or false? Give reasons for your answers.

e. arctan x = O(1)

112. True, or false? Give reasons for your answers.

a. 1/x⁴ = O(1/x² + 1/x⁴)

112. True, or false? Give reasons for your answers.

e. sec^(-1)x = O(1)

5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

g. 1/x

6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

a. log_2(x²)

6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

c. 1/√x

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

39. lim (x → ∞) (ln 2x - ln(x + 1))

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

44. lim (x → 0⁺) (csc x - cot x + cos x)

7. Order the following functions from slowest growing to fastest growing as x→∞.

a. e^x

b. x^x

c. (ln x)^x

d. e^(x/2)

17. Show that √(10x+1) and √(x+1) grow at the same rate as x→∞ by showing that they both grow at the same rate as √x as x→∞.

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

b. f(x)=x, g(x)=x + 1/x

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

e. f(x) = arccsc(x), g(x) = 1/x

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

a. f(x) = 3^(-x), g(x) = 2^(-x)