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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. COS X lim x-0 3x
Verify each limit using the limit definition. For example, in Exercise 13, show that |3x − 12| can be made as small as desired by taking x close to 4. lim(3x² - 9) = -9 x→0
Give examples of functions ƒ and g such that lim f(x) = lim g(x), but f(x) = g(x) for all x, including 0. x-0 x-0
Verify each limit using the limit definition. For example, in Exercise 13, show that |3x − 12| can be made as small as desired by taking x close to 4. lim(4x² + 2x + 5) = 5 x-0
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. 2x² - 9 x+3x²2x-3 lim
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. 1 lim x-4 (x-4)³
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. x²+x-6 x-2x²-x-2 lim
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. sin 3x lim x-0 3x
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. lim x→0 sin 2x X
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. sin x lim x+0 x²
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. lim x-0 sin 5x X
Sketch the graph of a function with the given limits. lim f(x) = ∞, x-1 ∞, lim f(x) = 0, lim f(x) = -00
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. 9-X+X E-X u E + X
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. 3-x lim x-1-x-1
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. lim 0→n/4 tan 0 - 2 sin 0 cos 0 0-л/4
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. lim(1 + 2r)¹/ T-0
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. x-4 lim x+3+ x²-9
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. x + 1 lim x-2x + 2
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. lim cos h→0 1 h
Determine the one-sided limits at c = 1, 2, and 4 of the function g shown in Figure 12, and state whether the limit exists at these points. 3 2 1 y 1 2 3 4 FIGURE 12 5 -X
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ∞ or −∞. lim |x|x x-0
The greatest integer function, also known as the floor function, is defined by where n is the unique integer such that n ≤ x . Calculate for c an integer: [x] = n
Sketch the graph of a function with the given limits. lim f(x) = f(2) = 3, X-2+ lim f(x) = -1, lim f(x) = 2 + f(4) x-2- X-4
Sketch the graph of a function with the given limits. lim f(x) = ∞o, 00, lim f(x) = 3, x-1- lim f(x)= =18 X-4
Sketch the graph of a function with the given limits. lim f(x) = 2, lim f(x) = 0, lim f(x) = 4 x-1 x-3- X-3+
Determine the one-sided limits at c = 2 and c = 4 of the function ƒ in Figure 13. What are the vertical asymptotes of ƒ? y 15+ 10+ 5+ -5 2 ----- -X
Plot the function and use the graph to estimate the value of the limit lim X-0 2x - cos x X
Determine the one-sided limits of the function ƒ in Figure 15, at the points c = 1, 3, 5, 6. 5 4 3 2 1 -1 -2 -3 -4 y 2 3 5 6 8 X
Plot the function and use the graph to estimate the value of the limit 12 - 1 lim x-0 4-1
Investigate numerically for several positive integer values of n. Then guess the value in general. sin no lim 0→0 0
Evaluate the limit using the Basic Limit Laws and the limits lim z2/3 Z-27
Plot the function and use the graph to estimate the value of the limit sin² 40 lim 0-0 cos 0 - 1
Plot the function and use the graph to estimate the value of the limit lim 8-0 sin² 20-0 sin 40
Does either of the two oscillating functions in Figure 16 appear to approach a limit as x → 0? AMA A (A) Hantex X AMAMA (B)
Plot the function and use the graph to estimate the value of the limit lim 0→0 cos 70 cos 50 0²
Evaluate the limit using the Basic Limit Laws and the limits lim xp/9 = c/a and lim k = k. X-C X-C
Plot the graph of (a) Zoom in on the graph to estimate(b) Explain why Use this to determine L to three decimal places. f(x) = 2 - 8 x - 3
In some cases, numerical investigations can be misleading. Plot ƒ(x) = cos π/x.(a) Does exist?(b) Show, by evaluating ƒ(x) at x = ±1/2, ±1/4 , ±1/6, . . . , that you might be able to trick your friends into believing that the limit exists and is equal to L = 1.(c) Which sequence of
Let for n a positive integer. Investigate L(n) numerically for several values of n, and then guess the value of L(n) in general. L(n) = lim n x→1 x-11-xn 1 1- x
Evaluate the limit using the Basic Limit Laws and the limits lim 1-25 3 √t-t (1-20)²
Light waves of frequency λ passing through a slit of width a produce a Fraunhofer diffraction pattern of light and dark fringes (Figure 17). The intensity as a function of the angle θ iswhere R = πa/λ and Im is a constant. Show that the intensity function is not defined at θ = 0. Then choose
The function is defined for x ≠ 0. f(x) = 2¹/x - 2-1/x 21/x + 2-1/x
Evaluate the limit using the Basic Limit Laws and the limits 3 لن lim 2-0 Z-1 >0
Find by numerical experimentation the positive integers k such that lim x→0 sin(sin² x) xk exists.
Show numerically that is less than 2 with b = 7 and is greater than 2 with b = 8. Experiment with values of b to find an approximate value of b for which the limit is 2. lim bt - 1 x X
Evaluate the limit using the Basic Limit Laws and the limits lim (3x³ + 2x²) X
Evaluate the limit using the Basic Limit Laws and the limits lim (16y + 1)(2y¹/2 + 1)
Evaluate the limit using the Basic Limit Laws and the limits limx-2 x→5
Evaluate the limit using the Basic Limit Laws and the limits 1 lim 1-4 t + 4
Evaluate the limit using the Basic Limit Laws and the limits lim (3x + 4) x-0.2
Evaluate the limit using the Basic Limit Laws and the limits lim (4x + 1)(6x - 1) I→
Evaluate the limit using the Basic Limit Laws and the limits lim (3x42x³+4x)
Evaluate the limit using the Basic Limit Laws and the limits - lim(x + 1)(3x2 - 9) X-2
Evaluate the limit using the Basic Limit Laws and the limits 3t - 14 lim 1-4 t +1
Evaluate the limit using the Basic Limit Laws and the limits lim(3x²/3 - 16x-¹) X-8
Evaluate the limit using the Basic Limit Laws and the limits √w +2+1 lim w 7 √w-3-1
Evaluate the limit using the Basic Limit Laws and the limits lim(18y² - 4)4
Evaluate the limit using the Basic Limit Laws and the limits 1 lim y→4 √бу + 1
Evaluate the limit using the Basic Limit Laws and the limits X lim x1 x³ + 4x
Evaluate the limit using the Basic Limit Laws and the limits √Z lim 2-9 Z-2
Evaluate the limit using the Basic Limit Laws and the limits 1² +1 lim 11 (1³ + 2)(t4 + 1)
Use Theorems 1–4 to show that the function is continuous.f (x) = 3x + 4 sin x THEOREM 1 Basic Laws of Continuity If f and g are continuous at x = c, then the following functions are also continuous at x = C: (i) f + g and f-g (iii) fg (ii) kf for any constant k (iv) f/g if g(c) #0
Evaluate the limit using the Basic Limit Laws and the limits lim x(x + 1)(x + 2) x→2
Assume that In each case evaluate the limit or indicate that the limit does not exist. if lim f(x) = L, then lim cos f(x) = cos L. x→a x→a
Suppose that Show that exists and equals 4. lim tg(t) = 12. 1-3
Use Theorems 1–4 to show that the function is continuous.ƒ(x) = x sin x THEOREM 1 Basic Laws of Continuity If f and g are continuous at x = c, then the following functions are also continuous at x = C: (i) f + g and f-g (iii) fg (ii) kf for any constant k (iv) f/g if g(c) #0
Evaluate the limit using the Basic Limit Laws and the limits (t + 2)¹/2 lim 1-7 (t + 1)2/3
Evaluate the limit using the Basic Limit Laws and the limits lim(4t² + 8t5)3/2 مانا
Use Theorems 1–4 to show that the function is continuous.ƒ(x) = x + sin x THEOREM 1 Basic Laws of Continuity If f and g are continuous at x = c, then the following functions are also continuous at x = C: (i) f + g and f-g (iii) fg (ii) kf for any constant k (iv) f/g if g(c) #0
Evaluate the limit assuming that lim f(x) = 3 and lim g(x) = 1. X-4 x-4
Evaluate the limit assuming that lim f(x) = 3 and lim g(x) = 1. X--4 X-4
Assuming that , compute: lim f(x) = 4 x→6
Show that the Product Law cannot be used to evaluate the limit lim (x-4) tan x. X→π/2
Assuming that which of the following statements is necessarily true? Why? lim f(x) →0 X || 1,
Suppose that (a) Explain why for any constant a ≠ 0.(b) If we assume instead that is it still necessarily true that (c) Illustrate (a) and (b) with the function ƒ(x) = x2. lim g(h) = L. h→0
Refer to the function g whose graph appears in Figure 16.State whether g is left- or right-continuous (or neither) at each of its points of discontinuity. 5 5 4 3- نها 2 लोक + + + 3 4 1 2 ( 5 + → X 6
Find the point c1 at which g has a jump discontinuity but is left-continuous. How should g(c1) be redefined to make g right-continuous at x = c1?
Assume that In each case evaluate the limit or indicate that the limit does not exist. if lim f(x) = L, then lim sin f(x) = sin L. x→a x→a
Suppose that ƒ(x) = 2 for x < 3 and ƒ(x) = −4 for x > 3.(a) What is ƒ(3) if ƒ is left-continuous at x = 3?(b) What is ƒ(3) if ƒ is right-continuous at x = 3?
Show that if both lim f(x) g(x) and lim g(x) exist and X-C X→C
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. k(x) = x-2 12-x|
Use Theorems 1–4 to show that the function is continuous.ƒ(x) = cos(x2) THEOREM 1 Basic Laws of Continuity If f and g are continuous at x = c, then the following functions are also continuous at x = C: (i) f + g and f-g (iii) fg (ii) kf for any constant k (iv) f/g if g(c) #0
Use Theorems 1–4 to show that the function is continuous. THEOREM 1 Basic Laws of Continuity If f and g are continuous at x = c, then the following functions are also continuous at x = C: (i) f + g and f-g (iii) fg (ii) kf for any constant k (iv) f/g if g(c) #0
Use Theorems 1–4 to show that the function is continuous.ƒ(x) = 3x3 + 8x2 − 20x THEOREM 1 Basic Laws of Continuity If f and g are continuous at x = c, then the following functions are also continuous at x = C: (i) f + g and f-g (iii) fg (ii) kf for any constant k (iv) f/g if g(c) #0
Use Theorems 1–4 to show that the function is continuous. THEOREM 1 Basic Laws of Continuity If f and g are continuous at x = c, then the following functions are also continuous at x = C: (i) f + g and f-g (iii) fg (ii) kf for any constant k (iv) f/g if g(c) #0
Use Theorems 1–4 to show that the function is continuous. THEOREM 1 Basic Laws of Continuity If f and g are continuous at x = c, then the following functions are also continuous at x = C: (i) f + g and f-g (iii) fg (ii) kf for any constant k (iv) f/g if g(c) #0
Use Theorems 1–4 to show that the function is continuous. THEOREM 1 Basic Laws of Continuity If f and g are continuous at x = c, then the following functions are also continuous at x = C: (i) f + g and f-g (iii) fg (ii) kf for any constant k (iv) f/g if g(c) #0
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. f(x)= = X
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. .h(x) = 1 2- |x|
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. f(x) = [x]
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. f(x) = x + 1 4x - 2
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. f(x) = x-2 |x-1|
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. |h(z) = 1-2z 7²-2-6
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. g(t) = 1 1² - 1
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous.ƒ(x) = |x|
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. f(x) = X L2
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. +1 w(t) = P-1
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. f(x) = COS 1 1 X X # 0 x = 0
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. x-2 f(x) = |x-21 -1 X #2 x = 2
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous.ƒ(x) = 3x2/3 − 9x3

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