1. Limits and Continuity

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 3x + 2) / (x³ − 4x) as

b. x→−2⁺

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.

lim (2 − 3 / t¹/³) as

a. t → 0⁺

Limits of quotients

Find the limits in Exercises 23–42.

limt→−2 (−2x − 4) / (x³ + 2x²)

Limits of quotients

Find the limits in Exercises 23–42.

limx→1 (x −1) / (√(x + 3) − 2)

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.)