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Problem 4.7.41
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (e¹/ₓ - 1)/(1/x)
Problem 4.7.42
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (tan⁻¹ x - π/2)/(1/x)
Problem 4.7.43
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ -1 (x³ - x² - 5x - 3)/(x⁴ + 2x³ - x² -4x -2)
Problem 4.7.46
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (ln(3x + 5eˣ)) / (ln(7x + 3e²ˣ)
Problem 4.7.47
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_v→3 (v-1-√(v²-5)) / (v-3)
Problem 4.7.48
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_y→2 (y²+y-6) / (√(8-y²)-y)
Problem 4.7.53
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 x csc x
Problem 4.7.54
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→1⁻ (1-x) tan πx/2
Problem 4.7.55
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 csc 6x sin 7x
Problem 4.7.57
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→π/2⁻ (π/2 - x) sec x
Problem 4.7.70
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (log₂ x - log₃ x)
Problem 4.7.72
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→π/2⁻ (π - 2x) tan x
Problem 4.7.73
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ x³ (1/x - sin 1/x)
Problem 4.7.75
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0⁺ x²ˣ
Problem 4.7.77
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_Θ→π/2⁻ (tan Θ)ᶜᵒˢ ᶿ
Problem 4.7.83
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 (x + cos x)¹/ˣ
Problem 4.7.31
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_u→ π/4 (tan u - cot u) / (u - π/4)
Problem 4.7.32
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_z→0 (tan 4z) / (tan 7z)
Problem 4.7.33
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (1 - cos 3x) / 8x²
Problem 4.7.34
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (sin² 3x) / x²
Problem 4.7.35
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→π (cos x +1 ) / (x - π )²
Problem 4.8.25
{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = ln x - x² + 3x - 1
Problem 4.8.37a
{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>
a. Find the time at which the object first passes the rest position, y = 0.
Problem 4.8.37b
{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>
b. Find the time and the displacement when the object reaches its lowest point.
Problem 4.8.37c
{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>
c. Find the time at which the object passes the rest position for the second time.
Problem 4.8.37d
{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>
d. Find the time and the displacement when the object reaches its high point for the second time.
Problem 4.8.39
{Use of Tech} Estimating roots The values of various roots can be approximated using Newton’s method. For example, to approximate the value of ³√10, we let x = ³√10 and cube both sides of the equation to obtain x³ = 10, or x³ - 10 = 0. Therefore, ³√10 is a root of p(x) = x³ - 10, which we can approximate by applying Newton’s method. Approximate each value of r by first finding a polynomial with integer coefficients that has a root r. Use an appropriate value of x₀ and stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding.
r = 7¹/⁴
Problem 4.8.43
{Use of Tech} Newton’s method and curve sketching Use Newton’s method to find approximate answers to the following questions.
Where is the first local minimum of f(x) = (cos x)/x on the interval (0,∞) located?
Problem 4.8.45
{Use of Tech} Newton’s method and curve sketching Use Newton’s method to find approximate answers to the following questions.
Where are the inflection points of f(x) = (9/5)x⁵ - (15/2)x⁴ + (7/3)x³ + 30x² + 1 located?
Problem 4.8.48
{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = x³ - 3x² + x + 1