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
> Is there a number that is exactly 1 more than its cube?
> For what values of x is g continuous? 0 if x is rational g(x) x if x is irrational
> For what values of x is f continuous? |0 if xis rational f(x) 1 if x is irrational
> a. Prove Theorem 4, part 3. b. Prove Theorem 4, part 5.
> Prove that cosine is a continuous function.
> To prove that sine is continuous, we need to show that limx→a sinx = sina for every real number a. By Exercise 63 an equivalent statement is that Use (6) to show that this is true. lim sin(a + h) = sin a
> Prove that f is continuous at a if and only if lim f(a + h) = f(a) h→0
> Prove, without graphing, that the graph of the function has at least two xintercepts in the specified interval. y = x2 - 3 + 1/x, (0, 2)
> Prove, without graphing, that the graph of the function has at least two xintercepts in the specified interval. y = sin x3, (1, 2)
> a. Prove that the equation has at least one real root. b. Use your graphing device to find the root correct to three decimal places. arctan x = 1 - x
> Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. In x – In 4 lim X - 4
> a. Prove that the equation has at least one real root. b. Use your graphing device to find the root correct to three decimal places. 100e-x/100 = 0.01x2
> 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. ln x = 3 - 2x
> 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) = 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. g(t) = t2 + 5t/2t + 1, a = 2
> 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
