Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = 4 + 3x² / (x² + 1)

Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–20.

limx→1− (1/(x + 1))((x + 6)/x)((3 − x)/7)

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)

lim x → ±∞ f(x) = 0, lim x → 2⁻ f(x) = ∞, and lim x → 2⁺ f(x) = ∞

Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

x³ − 15x + 1 = 0 (three roots)

Theory and Examples

If limx→4 (f(x) − 5) / (x − 2) = 1, find limx→4 f(x).

Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = 2x / (x² − 1)

Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = 1/x, x = -2

Finding Limits

In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.

lim (4g(x))¹/³ = 2

x →0

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))

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→3 (3x − 7) = 2

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limt→0 2t / tan t

Theory and Examples

Once you know limx→a+ f(x) and limx→a− f(x) at an interior point of the domain of f, do you then know limx→a f(x)? Give reasons for your answer.

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limt→0 sin(1 − cos t) / (1 − cos t)

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx → √3 1/x² = 1/3

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 √(4 − x) = 2

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→−π √(x + 4) cos(x + π)

At what points are the functions in Exercises 13–30 continuous?

g(x) = { (x² − x – 6)/(x – 3), x ≠ 3

5, x = 3

Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–20.

limh→0− (√6 − √(5h² + 11h + 6))/ h

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0 sin(sin h) / sin h

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (1 − cos 3x) / 2x

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?

lim x → π/6 √(csc² x + 5√3 tan x)

Limits of quotients

Find the limits in Exercises 23–42.

limx→−1 (√(x² + 8) − 3) / (x + 1)

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→1 1/x = 1

At what points are the functions in Exercises 13–30 continuous?

y = 1/(x – 2) – 3x

Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = x², x = -2

Finding Limits of Differences When x → ±∞

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

lim x → −∞ (√(x² + 3) + x)

A function value Show that the function F(x) = ( x − a)²(x − b)² + x takes on the value (a + b)² for some value of x.

Calculating Limits

Find the limits in Exercises 11–22.

limt→6 8(t−5)(t−7)

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?

lim ϴ → 0 cos (πϴ/sin ϴ)

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 4 / x²/⁵