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Question: Sketch the graph of an example of

Sketch the graph of an example of a function f that satisfies all of the given conditions.
Sketch the graph of an example of a function f that satisfies all of the given conditions.





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lim f(x) = 3, lim f(x) = ∞, lim f(x) -00, ƒ is odd x→2- x→2+


> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y = 2ex/ ex - 5

> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y = x3 – x/x2 - 6x + 5

> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y = 1 + x4/x2 - x4

> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y = 2x2 + x – 1/x2 + x - 2

> a. By graphing the function f(x) = (cos 2x - cos x)/x2 and zooming in toward the point where the graph crosses the y-axis, estimate the value of limx→0 f(x). b. Check your answer in part (a) by evaluating f(x) for values of x that approach 0.

> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y = 2x2 + 1/3x2 + 2x - 1

> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y = 5 + 4x/x + 3

> 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 – 1/ln x find each of the following limits. e. Use the information from parts (a)–(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) - 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.

> 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 (-&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") ")

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