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Problem 2.4.45
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limx→0 (1 − cos 3x) / 2x
Problem 2.2.17
Calculating Limits
Find the limits in Exercises 11–22.
limx→−1/2 4x(3x+4)²
Problem 2.6.77
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) = ∞
Problem 2.5.72
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)
Problem 2.1.6
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
P(θ)=θ³ − 4θ² + 5θ; [1,2]
Problem 2.6.70
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)
Problem 2.2.60
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
Problem 2.25
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
Problem 2.6.85
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))
Problem 2.3.5
Centering Intervals About a Point
In Exercises 1–6, sketch the interval (a,b), on the x-axis with the point c inside. Then find a value of δ>0 such that a < x < b whenever 0 < |x−c| < δ.
a=4/9, b=4/7, c=1/2
Problem 2.4.47
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.
Problem 2.5.29
At what points are the functions in Exercises 13–30 continuous?
g(x) = { (x² − x – 6)/(x – 3), x ≠ 3
5, x = 3
Problem 2.4.28
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limt→0 2t / tan t
Problem 2.2.58
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
Problem 2.4.36
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limh→0 sin(sin h) / sin h
Problem 2.46
Limits and Infinity
Find the limits in Exercises 37–46.
x²/³ + x⁻¹
lim --------------------
x→∞ x²/³ + cos²x
Problem 2.4.40
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 sin θ cot 2θ
Problem 2.4.16
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
limh→0− (√6 − √(5h² + 11h + 6))/ h
Problem 2.3.37
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx →4 (9 − x) = 5
Problem 2.5.40
Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?
lim ϴ → 0 cos (πϴ/sin ϴ)
Problem 2.3.43
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
lim x→1 1/x = 1
Problem 2.5.27
At what points are the functions in Exercises 13–30 continuous?
y = (2x – 1)¹/³
Problem 2.6.27
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→∞ (2√x + x⁻¹) / (3x − 7)
Problem 2.5.13
At what points are the functions in Exercises 13–30 continuous?
y = 1/(x – 2) – 3x
Problem 2.3.38
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx→3 (3x − 7) = 2
Problem 2.6.86
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)
Problem 2.5.54
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.
Problem 2.2.13
Calculating Limits
Find the limits in Exercises 11–22.
limt→6 8(t−5)(t−7)
Problem 2.2.59
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
Problem 2.2.77
Theory and Examples
If limx→4 (f(x) − 5) / (x − 2) = 1, find limx→4 f(x).