1. Limits and Continuity

82. Use the definitions of the hyperbolic functions to find each of the following limits.

h. lim(x→0^-) coth x

9. True, or false? As x→∞,

g. ln(x) = o(ln(2x))

82. Use the definitions of the hyperbolic functions to find each of the following limits.

i. lim(x→-∞) csch x

10. True, or false? As x→∞,

a. 1/(x+3) = O(1/x)

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)