1. Limits and Continuity

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→0 (2sin x − 1)

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→0 √(7 + sec²x)

Oblique Asymptotes

Graph the rational functions in Exercises 103–108. Include the graphs and equations of the asymptotes.

y = x² / (x − 1)

Oblique Asymptotes

Graph the rational functions in Exercises 103–108. Include the graphs and equations of the asymptotes.

y = (x² − 4) / (x − 1)

Oblique Asymptotes

Graph the rational functions in Exercises 103–108. Include the graphs and equations of the asymptotes.

y = (x² − 1) / x

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→0 (1 + x + sin x) / (3 cosx)

Additional Graphing Exercises

[Technology Exercise] Graph the curves in Exercises 109–112. Explain the relationship between the curve’s formula and what you see.

y = −1 / √(4 − x²)

Finding Limits of Differences When x → ±∞

Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)

lim x → ∞ (√(x² + 25) − √(x² − 1))

Limit Evaluate lim x → ∞ (tanh x)ˣ.

Behavior at the origin Using calculus and accurate sketches, explain how the graphs of f(x) = xᵖ ln x differ as x → 0⁺ for p = 1/2, 1, and 2.

Log-normal probability distribution A commonly used distribution in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is

f(x) = 1/xσ√(2π) e⁻ˡⁿ^² ˣ / ²σ^², for x ≥ 0

where ln x has zero mean and standard deviation σ > 0.

b. Evaluate lim x → 0 ƒ(x). (Hint: Let x = eʸ.)

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)