1. Limits and Continuity

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

69. lim (x → ∞) (√(9x + 1)) / (√(x + 1))

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

71. lim (x → (π/2)⁻) sec x / tan x

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

73. lim (x → ∞) (2^x - 3^x) / (3^x + 4^x)

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

75. lim (x → ∞) e^(x²) / (x e^x)

84. Find lim(x→∞) (√(x² + 1) - √x).

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

13. lim(x → 1⁻)arcsin(x)

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

15. lim(x→∞)arctan(x)

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

17. lim(x→∞)arcsec(x)

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

19. lim(x→∞)arccsc(x)

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

a. lim(x→∞) tanh x

9. True, or false? As x→∞,

a. x = o(x)

9. True, or false? As x→∞,

c. x = O(x+5)

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

c. lim(x→∞) sinh x

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

f. lim(x→∞) coth x

9. True, or false? As x→∞,

e. e^x = o(e^(2x))