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Question: To prove that sine is continuous, we

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
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.

Use (6) to show that this is true.





Transcribed Image Text:

lim sin(a + h) = sin a


> Find the limit or show that it does not exist. sin?x 2. lim .2 + 1

> Find the limit or show that it does not exist. 1 — е* lim x0 1 + 2e*

> Find the limit or show that it does not exist. e t - e lim x0 e 3x + e -3x — е

> Find the limit or show that it does not exist. lim arctan(e*)

> Find the limit or show that it does not exist. 1 + x* lim x* + 1 x-0

> Find the limit or show that it does not exist. lim (x² + 2x²) X-0

> Find the limit or show that it does not exist. lim (e* + 2 cos 3x)

> Find the limit or show that it does not exist. x* – 3x? + x lim x + 2

> Find the limit or show that it does not exist. lim Vx2 + 1

> Find the limit or show that it does not exist. lim (Vx2 + ax Vx² + bx

> 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. lim x*

> Find the limit or show that it does not exist. lim (V4x2 + 3x + 2x)

> Find the limit or show that it does not exist. lim (/9x? + x – 3x)

> Find the limit or show that it does not exist. x + 3x² lim x0 4x - 1 .2

> Find the limit or show that it does not exist. Vx + 3x? lim 4х — 1

> Find the limit or show that it does not exist. V1 + 4x6 lim .3 X -0 2 - x

> Find the limit or show that it does not exist. V1 + 4x6 lim .3 2 -

> Find the limit or show that it does not exist. x2 lim /x* + 1

> Find the limit or show that it does not exist. (2x² + 1)² lim (x – 1)(x² + x)

> Find the limit or show that it does not exist. lim a 2t3/2 + 3t – 5

> Find the limit or show that it does not exist. VE + t? lim 2t – t?

> 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. 5' – 1 lim 0 t

> Find the limit or show that it does not exist. 4x3 + 6х2 — 2 lim 2x3 — 4х + 5 X-00

> Find the limit or show that it does not exist. x - 2 lim x-0 x2 + 1

> Find the limit or show that it does not exist. 1 - x? .2 lim .3 x→* x' - x + 1

> Find the limit or show that it does not exist. Зх — 2 lim х 2х + 3x >00

> Evaluate the limit and justify each step by indicating the appropriate properties of limits. 9x³ + 8x — 4 lim 3 — 5х + x3

> Evaluate the limit and justify each step by indicating the appropriate properties of limits. 2x? – 7 lim 5x? + x – 3

> a. Use a graph of f(x) = (1 – 2/x)2 to estimate the value of limx→∞ f(x) correct to two decimal places. b. Use a table of values of f(x) to estimate the limit to four decimal places.

> Guess the value of the limit by evaluating the function f(x) = x2/2x for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100. Then use a graph of f to support your guess. .2 lim X→* 2*

> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) lim f(x) = 2, f(0) = 0, f is even -00, x→3

> Sketch the graph of an example of a function f that satisfies all of the given conditions. f(0) = 3, lim f(x) = 4, x>0- lim f(x) = 2, x→0+ lim f(x) lim f(x) x→4- lim f(x) -00, -00, 00, X -00 X→4+ lim f(x) = 3

> 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. sin 30 S lim 00 tan 20

> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) = 3, lim f(x) = ∞, lim f(x) -00, ƒ is odd x→2- x→2+

> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) - 0o, lim f(x) = ∞, lim f(x) = 0, x→2 X00 X -00 lim f(x) x0+ / (X) :

> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) = ∞, x→2 lim f(x) = ∞, x→-2+ lim f(x) = x→-2- im J(x) = 0, lim f(x) = 0, f(0) = 0 X -00

> a. From the graph of f, state the numbers at which f is discontinuous and explain why. b. For each of the numbers stated in part (a), determine whether f is continuous from the right, or from the left, or neither. y A -2 4 6. 2.

> If f is continuous on (-∞, ∞), what can you say about its graph?

> Write an equation that expresses the fact that a function f is continuous at the number 4.

> A Tibetan monk leaves the monastery at 7:00 am and takes his usual path to the top of the mountain, arriving at 7:00 pm. The following morning, he starts at 7:00 am at the top and takes the same path back, arriving at the monastery at 7:00 pm. Use the In

> a. Show that the absolute value function F(x) = |x| is continuous everywhere. b. Prove that if f is a continuous function on an interval, then so is | f |. c. Is the converse of the statement in part (b) also true? In other words, if | f | is continuou

> Show that the function is continuous on (-∞, ∞) x* f(x) = sin(1/x) if x 0 if x = 0

> If a and b are positive numbers, prove that the equation a/ x3 + 2x2 – 1 + b/ x3 + x - 2 = 0 has at least one solution in the interval (-1, 1).

> 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. 1 + p° lim p1 1+ p5

> 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.

> 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 (-&acirc;&#136;&#158;, &acirc;&#136;&#158;)? 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 (-&acirc;&#136;&#158;, &acirc;&#136;&#158;). 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 (-&acirc;&#136;&#158;, &acirc;&#136;&#158;). 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

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