Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

llimx→0 (x −x cos x) / sin² 3x

Limits of quotients

Find the limits in Exercises 23–42.

limu→1 (u⁴ − 1)/(u³ − 1)

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 → ±∞ k(x) = 1, lim x → 1⁻ k(x) = ∞, and lim x → 1⁺ k(x) = −∞

Limits and Infinity

Find the limits in Exercises 37–46.

x²/³ + x⁻¹

lim --------------------

x→∞ x²/³ + cos²x

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→−π √(x + 4) cos(x + π)

In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?

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Calculating Limits

Find the limits in Exercises 11–22.

limx→−1/2 4x(3x+4)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 + 25) − √(x² − 1))

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)

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)

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

y = √(x⁴ +1)/(1 + sin² x)

Horizontal and Vertical Asymptotes

Determine the domain and range of y = (√16―x²) / (x―2).

Limits of quotients

Find the limits in Exercises 23–42.

limx→−3 (2 − √(x² − 5)) / (x + 3)

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (1 − cos 3x) / 2x

Infinite Limits

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

lim x→0 4 / x²/⁵

Finding Limits of Differences When x → ±∞

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

lim x → ∞ (√(9x² − x) − 3x)

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→−3 (x² − 9) / (x + 3) = −6

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

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) = 3x - 4, x = 2

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 x sin (1/x) = 0

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

lim x → 0 tan (π/4 cos (sin x¹/³))

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

lim ϴ → 0 cos (πϴ/sin ϴ)

Limits and Infinity

Find the limits in Exercises 37–46.

sin x

lim ------------- ( If you have a grapher, try graphing

x→∞ |x| the function for ―5 ≤ x ≤ 5 ) .

Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x→∞ (x − 3) / √(4x² + 25)

Finding Limits of Differences When x → ±∞

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

lim x → −∞ (2x + √(4x² + 3x − 2))

Horizontal and Vertical Asymptotes

Assume that constants a and b are positive. Find equations for all horizontal and vertical asymptotes for the graph of y = (√ax² + 4) / (x―b) .

Suppose that a function f(x) is defined for all x in [-1,1]. Can anything be said about the existence of limx→0 f(x)? Give reasons for your answer.

Theory and Examples

Suppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.

Use formal definitions to prove the limit statements in Exercises 93–96.

lim x → −5 (1 / (x + 5)²) = ∞

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.