Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. g(t) = t2 + 5t/2t + 1, a = 2
> a. Prove that the equation has at least one real root. b. Use your calculator to find an interval of length 0.01 that contains a root. cos x = x3
> Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. sin x = x2 - x, (1, 2)
> Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. ex = 3 - 2x, (0, 1)
> Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. ln x = x -√x , (2, 3)
> Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. x4 + x - 3 = 0, (1, 2)
> Suppose f is continuous on [1, 5] and the only solutions of the equation f(x) = 6 are x = 1 and x = 4. If f(2) = 8, explain why f(3) > 6.
> If f(x) = x2 + 10 sin x, show that there is a number c such that f(c) = 1000.
> Suppose that a function f is continuous on [0, 1] except at 0.25 and that f(0) = 1 and f(1) = 3. Let N = 2. Sketch two possible graphs of f, one showing that f might not satisfy the conclusion of the Intermediate Value Theorem and one showing that f migh
> Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). h = (0.5, (0.1, (0.01, (0.001, (0.0001 (2 + h) – 32 lim h
> Which of the following functions f has a removable discontinuity at a? If the discontinuity is removable, find a function t that agrees with f for x ≠ a and is continuous at a. a. f(x) = x4 – 1/ x-1, a = 1 b. f(x) = x3 –x2 – 2x/ x- 2, a = 2 c. f(x) = [[s
> Let f(x) = 1/x and g(x) = 1/x2. a. Find (f o g)(x). b. Is f + g continuous everywhere? Explain.
> Suppose f and g are continuous functions such that g(2) = 6 and limx→2 [3f(x) + f(x)g(x)] = 36. Find f(2).
> Find the values of a and b that make f continuous everywhere. x? – 4 if x< 2 x - 2 f(x) = ax? – bx + 3 if 2<x<3 2х — а + b if x>3
> For what value of the constant c is the function f continuous on (-∞, ∞)? Scx? + 2x if x < 2 f(x) = Cx if x> 2
> The gravitational force exerted by the planet Earth on a unit mass at a distance r from the center of the planet is where M is the mass of Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r? GMr if r<R R3 F
> Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph off. х+ 2 if x <0 if 0<x<1 f(x) = {e* 2 — х if x> 1
> Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph off. 2* if x<1 f(x) = {3 – x if 1<x< 4 Vx if x>4
> Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph off. x2 if -1 <x<1 if x<-1 f(x) = 1/x if x> 1
> Show that f is continuous on (-∞, ∞). sin x if x < T/4 cos x if x > T/4 S(x) =
> Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). t = (0.5, (0.1, (0.01, (0.001, (0.0001 est – 1 lim
> Show that f is continuous on (-∞, ∞). 1 - x? if x < 1 if x>1 f(x) = Inx
> Use continuity to evaluate the limit. lim 3- →4 2x-4 X-
> Use continuity to evaluate the limit. 5 – x? lim In 1 + x
> Use continuity to evaluate the limit. lim sin(x + sin x)
> Use continuity to evaluate the limit. lim x /20 – x' .2
> Locate the discontinuities of the function and illustrate by graphing. y = ln(tan2x)
> Locate the discontinuities of the function and illustrate by graphing. y = 1 / 1 + e1/x
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. N(r) = tan'(1 +e¯r") ")
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. M(x) = 1 + 1/x
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. B(x) = tan x/ 4 − x2
> Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). x = -2.5, -2.9, -2.95, -2.99, -2.999, -2.9999, -3.5, -3.1, -3.05, -3.01, -3.001, -3.0001 x? – 3x lim 3 x? - 9 ' 3
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. A(t) = arcsin (1 +2t)
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. R(t) = e sin t/ 2 + cos π t
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. Q(x) = 3 x – 2 / x3 - 2
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. G(x) = x2 + 1/ 2x2 – x - 1
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. f(x) = 2x2 –x – 1/ x2 + 1
> How would you “remove the discontinuity” off? In other words, how would you define fs2d in order to make f continuous at 2? f(x) = x3 – 8/x2 - 4
> How would you “remove the discontinuity” off? In other words, how would you define fs2d in order to make f continuous at 2? f(x) = x2 - x – 2/x - 2
> Explain why the function is discontinuous at the given number a. Sketch the graph of the function. 2x? – 5x – 3 - if x + 3 f(x) = x – 3 a = 3 if x = 3
> Explain why the function is discontinuous at the given number a. Sketch the graph of the function. cos x if x <0 f(x) = if x = 0 a = 0 1 - x? if x > 0
> Explain why the function is discontinuous at the given number a. Sketch the graph of the function. x2 if x + 1 .2 f(x) = - 1 a = 1 1 if x = 1
> Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). x = 3.1, 3.05, 3.01, 3.001, 3.0001, 2.9, 2.95, 2.99, 2.999, 2.9999 x2 - 3x lim 3 x - 9' — Зх
> Explain why the function is discontinuous at the given number a. Sketch the graph of the function. |x + 3 if x < -1 f(x) = 2* a = -1 if x> -1
> Explain why the function is discontinuous at the given number a. Sketch the graph of the function. 1 if x + -2 f(x) x + 2 a = -2 if x = -2
> Explain why the function is discontinuous at the given number a. Sketch the graph of the function. f(x) = 1/x + 2 a = -2
> Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. g(x) = x – 1/3x + 6, (-∞, -2)
> Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. f(x) = x + x − 4 , (4, ∞)
> Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. f(x) = 3x4 - 5x + 3 x2 + 4 , a = 2
> Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. p(v) = 2 3v2 + 1 , a = 1
> Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. f(x) = (x + 2x3)4, a = -1
> Explain why each function is continuous or discontinuous. a. The temperature at a specific location as a function of time b. The temperature at a specific time as a function of the distance due west from New York City c. The altitude above sea level as
> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) = 2, lim f(x) = 0, lim f(x)= 3, lim f(x) = 0, f(0) = 2, f(4) = 1
> The toll T charged for driving on a certain stretch of a toll road is $5 except during rush hours (between 7 am and 10 am and between 4 pm and 7 pm) when the toll is $7. a. Sketch a graph of T as a function of the time t, measured in hours past midnigh
> Sketch the graph of a function f that is continuous except for the stated discontinuity. Neither left nor right continuous at -2, continuous only from the left at 2
> Sketch the graph of a function f that is continuous except for the stated discontinuity. Removable discontinuity at 3, jump discontinuity at 5
> Sketch the graph of a function f that is continuous except for the stated discontinuity. Discontinuities at -1 and 4, but continuous from the left at -1 and from the right at 4
> Sketch the graph of a function f that is continuous except for the stated discontinuity. Discontinuous, but continuous from the right, at 2
> From the graph of g, state the intervals on which g is continuous. -3 -2 1 2 3.
> Use the given graph of f(x) = x2 to find a number δ such that if |x - 1| 1.5- y =x? 1 0.5 ? ?
> Use the given graph of f(x) = √x to find a number such that if |x - 4| δ then | x − 2 | yA ソ=Vx 2.4 2 1.6 ? 4 ?
> Use the given graph off to find a number δ such that if 0 yA 2.5 2- 1.5 2.6 3 3.8
> Use the given graph of f to find a number δ such that if |x - 1| yA 1.2 1 0.8 0.7 1 1.1
> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) = 4, lim f(x) = 2, lim f(x) = 2, X-2 f(3) = 3, f(-2) = 1
> Suppose that limx→a f(x) = ∞ and limx→a g(x) = c, where c is a real number. Prove each statement. (а) lim [f(x) + glx)] — 00 (b) lim [/(x)g(x)] — оо if c > 0 (с) lim [f(x)g(х)] —D — оо if c <0
> Prove that / ln x = -∞.
> Prove, using Definition 6, that 1 lim (x + 3)* 00 X-3
> How close to -3 do we have to take x so that 1 > 10,000 (x + 3)4
> By comparing Definitions 2, 3, and 4, prove Theorem 2.3.1.
> If the function f is defined by prove that limx→0 f(x) does not exist. 0 if x is rational 1 if x is irrational f(x) =
> If H is the Heaviside function defined in Example 2.2.6, prove, using Definition 2, that limt→0 H(t) does not exist.
> Prove that / √x = √a if a > 0.
> Prove that 1 lim x→2 X 2
> a. For the limit limx → 1 (x3 + x + 1) = 3, use a graph to find a value of δ that corresponds to ε = 0.4. b. By using a computer algebra system to solve the cubic equation x3 + x + 1 = 3 + ε, find the largest possible value of δ that works for any giv
> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) = 1, lim f(x)= -2, lim f(x) = 2, S(0) = -1, f(3) = 1
> Verify, by a geometric argument, that the largest possible choice of δ for showing that limx→3 x2 = 9 is δ = 9 + ε - 3.
> Verify that another possible choice of δ for showing that limx→3 x2 = 9 in Example 4 is δ = min{2, ε/8}.
> Prove the statement using the ε, δ definition of a limit. lim x3 = 8
> Prove the statement using the ε, δ definition of a limit. lim (x2 – 1) = 3 X-2
> Prove the statement using the ε, δ definition of a limit. lim (x? + 2x – 7) = 1 %3D
> Prove the statement using the ε, δ definition of a limit. lim (x? – 4x + 5) = 1
> Prove the statement using the ε, δ definition of a limit. lim 16 + x = 0 X→-6+
> Prove the statement using the ε, δ definition of a limit. lim |x| = 0
> Prove the statement using the ε, δ definition of a limit. lim x = 0 .3
> Prove the statement using the ε, δ definition of a limit. lim x? = 0
> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) = -1, lim f(x) = 2, f(0) = 1 X0+ %3D
> Prove the statement using the ε, δ definition of a limit. lim c = c
> Prove the statement using the ε, δ definition of a limit. lim x = a
> Prove the statement using the ε, δ definition of a limit. 9 – 4x2 lim X→-1.5 3 + 2x
> Prove the statement using the ε, δ definition of a limit. x2 lim — 2х — 8 х — 4
> Prove the statement using the ε, δ definition of a limit. (3 – x) = -5 X10
> Prove the statement using the ε, δ definition of a limit. 2 + 4x = 2 3 lim
> Prove the statement using the ε, δ definition of a limit and illustrate with a diagram like Figure 9. From Figure 9 lim (3x + 5) = -1 X-2 y 4 y=4x- 5 7+8 1-8 3 3-8 3+8
> Prove the statement using the ε, δ definition of a limit and illustrate with a diagram like Figure 9. From Figure 9 lim (1 – 4x) = 13 x-3 y 4 y=4x- 5 7+8 1-8 3 3-8 3+8
> Prove the statement using the ε, δ definition of a limit and illustrate with a diagram like Figure 9. From Figure 9 lim (2x – 5) = 3 %3D y 4 y=4x- 5 7+8 1-8 3 3-8 3+8
> Prove the statement using the ε, δ definition of a limit and illustrate with a diagram like Figure 9. From Figure 9 lim (1 + x) = 2 y 4 y=4x- 5 7+8 1-8 3 3-8 3+8
> Use the graph of the function f to state the value of each limit, if it exists. If it does not exist, explain why. f(x) = x2 + x / x3 + x2 (a) lim f(x) (b) lim f(x) (c) lim f(x)
> Given that limx → 2 (5x – 7) = 3, illustrate Definition 2 by finding values of that correspond to ε = 0.1, ε = 0.05, and ε = 0.01.
> a. Find a number δ such that if |x - 2| < δ , then |4x - 8|< ε, where ε = 0.1. b. Repeat part (a) with ε = 0.01.
> A crystal growth furnace is used in research to determine how best to manufacture crystals used in electronic components for the space shuttle. For proper growth of the crystal, the temperature must be controlled accurately by adjusting the input power.
> A machinist is required to manufacture a circular metal disk with area 1000 cm2. a. What radius produces such a disk? b. If the machinist is allowed an error tolerance of (5 cm2 in the area of the disk, how close to the ideal radius in part (a) must th
