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
Use (6) to show that this is true.
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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 (-∞, ∞)? 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
