1.99
See Answer

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 continuous, does it follow that f is continuous? If so, prove it. If not, find a counterexample.

** >** a. Use a graph of f(x) = 3x2 + 8x + 6 - 3x2 + 3x + 1 to estimate the value of limx→∞ f(x) to one decimal place. b. Use a table of values of f(x) to estimate the limit to four decimal places. c. Find the exact value of the limit.

** >** a. Estimate the value of by graphing the function f(x) = x2 + x + 1 + x. b. Use a table of values of f(x) to guess the value of the limit. c. Prove that your guess is correct.
lim (Vx² + x + 1 + x) X -0

** >** For f(x) = 2/x â€“ 1/ln x find each of the following limits. e. Use the information from parts (a)â€“(d) to make a rough sketch of the graph of f.
(a) lim f(x) (b) lim f(x) X→0+ (c) lim f(x) (d) lim f(x) x→1- x→1+

** >** a. For f(x) = x/ln x find each of the following limits. b. Use a table of values to estimate c. Use the information from parts (a) and (b) to make a rough sketch of the graph of f.
(i) lim f(x) (ii) lim f(x) (iii) lim f(x) X→0+ I→I+ lim f(x).

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

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

** >** Find the limit or show that it does not exist.
lim tan (In x)

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

** >** 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' Inx -2

** >** 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

** >** Show that the function is continuous on (-âˆž, âˆž)
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.

** >** To prove that sine is continuous, we need to show that limxâ†’a sinx = sina for every real number a. By ExerciseÂ 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 (-âˆž, âˆž)?
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 (-âˆž, âˆž).
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 (-âˆž, âˆž).
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