1.99 See Answer

Question: For the limit / illustrate Definition 2 by

For the limit
For the limit


illustrate Definition 2 by finding values of  that correspond to ε = 0.5 and ε = 0.1.

illustrate Definition 2 by finding values of that correspond to ε = 0.5 and ε = 0.1.





Transcribed Image Text:

e 2x lim 1 = 2


> 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&acirc;&#134;&#146;a f(x) = &acirc;&#136;&#158; and limx&acirc;&#134;&#146;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&acirc;&#134;&#146;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 &Icirc;&micro;, &Icirc;&acute; definition of a limit. lim x3 = 8

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit. lim (x2 – 1) = 3 X-2

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit. lim (x? + 2x – 7) = 1 %3D

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit. lim (x? – 4x + 5) = 1

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit. lim 16 + x = 0 X→-6+

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit. lim |x| = 0

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit. lim x = 0 .3

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; 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 &Icirc;&micro;, &Icirc;&acute; definition of a limit. lim c = c

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit. lim x = a

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit. 9 – 4x2 lim X→-1.5 3 + 2x

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit. x2 lim — 2х — 8 х — 4

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit. (3 – x) = -5 X10

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit. 2 + 4x = 2 3 lim

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit and illustrate with a diagram like Figure 9. From Figure 9 lim (3x + 5) = -1 X-2 y 4 y=4x- 5 7+8 1-8 3 3-8 3+8

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit and illustrate with a diagram like Figure 9. From Figure 9 lim (1 – 4x) = 13 x-3 y 4 y=4x- 5 7+8 1-8 3 3-8 3+8

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit and illustrate with a diagram like Figure 9. From Figure 9 lim (2x – 5) = 3 %3D y 4 y=4x- 5 7+8 1-8 3 3-8 3+8

> Prove the statement using the &Icirc;&micro;, &Icirc;&acute; definition of a limit and illustrate with a diagram like Figure 9. From Figure 9 lim (1 + x) = 2 y 4 y=4x- 5 7+8 1-8 3 3-8 3+8

> Use the graph of the function f to state the value of each limit, if it exists. If it does not exist, explain why. f(x) = x2 + x / x3 + x2 (a) lim f(x) (b) lim f(x) (c) lim f(x)

> Given that limx → 2 (5x – 7) = 3, illustrate Definition 2 by finding values of  that correspond to ε = 0.1, ε = 0.05, and ε = 0.01.

> a. Find a number δ such that if |x - 2| < δ, then |4x - 8|< ε, where ε = 0.1. b. Repeat part (a) with ε = 0.01.

> A crystal growth furnace is used in research to determine how best to manufacture crystals used in electronic components for the space shuttle. For proper growth of the crystal, the temperature must be controlled accurately by adjusting the input power.

> A machinist is required to manufacture a circular metal disk with area 1000 cm2. a. What radius produces such a disk? b. If the machinist is allowed an error tolerance of (5 cm2 in the area of the disk, how close to the ideal radius in part (a) must th

> Given that limx →π csc2 x = ∞, illustrate Definition 6 by finding values of that correspond to a. M = 500 and b. M = 1000.

> a. Use a graph to find a number such that if 2 < x < 2 + δ then 1/ln(x – 1) > 100 b. What limit does part (a) suggest is true?

> For the limit illustrate Definition 2 by finding values of that correspond to &Icirc;&micro;= 0.2 and &Icirc;&micro; = 0.1. lim (x – 3x + 4) = 6

> Use a graph to find a number δ such that if |x - 1| < δ then | 2x/x2 + 4 - 0.4| < 0.1

> Use a graph to find a number δ such that if |x - π/4 | < δ then |tan x - 1| < 0.2

> Use the graph of the function f to state the value of each limit, if it exists. If it does not exist, explain why. f(x) = 1 / 1 + e1/y (a) lim f(x) (b) lim f(x) (c) lim f(x)

> The graphs of f and t are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why. (a) lim [f(x) + g(x)] (b) lim [f(x) – g(x)] (c) lim [f(x)g(x)] f(x) (d) lim X3 g(x) X-1 (e) lim [r²f(x)] (f) f(-1) + lim g(x)

> Given that find the limits that exist. If the limit does not exist, explain why. lim f(x) = 4 lim g(x) = -2 lim h(x) = 0 (a) lim [f(x) + 5g(x)] (b) lim [g(x)]³ X2 3f (x) (c) lim f(x) (d) lim 2 g(x) g(x) (e) lim 2 h(x) g(x)h(x) (f) lim 2 f(x)

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> Show by means of an example that limx→a [f(x) = g(x)] may exist even though neither limx→ a f(x) nor limx → a g(x) exists.

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> If p is a polynomial, show that lim x→a p(x) = p(a).

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> Let f(x) = [[cos x]], -&Iuml;&#128; &acirc;&#137;&curren; x &acirc;&#137;&curren; &Iuml;&#128; . a. Sketch the graph off. b. Evaluate each limit, if it exists. c. For what values of a does limx &acirc;&#134;&#146; a f(x) exist? (i) lim f(x) (ii)

> a. If the symbol [[ ]] denotes the greatest integer function defined in Example 10, evaluate b. If n is an integer, evaluate c. For what values of a does limx &acirc;&#134;&#146; a [[x]] exist? (i) lim. [x] (ii) lim [x] (iii) lim [x] -2+ -2 X-2.4

> Let a. Evaluate each of the following, if it exists. b. Sketch the graph of t. if x<1 3 g(x) : if x = 1 2 — х? х — 3 if 1<x<2 if x>2 (i) lim g(x) (ii) lim g(x) (iii) g(1) (iv) lim g(x) (v) lim g(x) (vi) lim g(x) 2+

> Let Find the value of c so that / exists. if t<2 Vi+c ift> 2 if t> 2 [4 - 31 B(t)

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> Let g(x) = x2 + x &acirc;&#128;&#147; 6/|x - 2| a. Find b. Does limx &acirc;&#134;&#146; 2 g(x) exist? c. Sketch the graph of t. (i) lim g(x) (ii) lim g(x) X→2+

> Let g(x) = sgn(sinx). a. Find each of the following limits or explain why it does not exist. b. For which values of a does limx &acirc;&#134;&#146; a g(x) not exist? c. Sketch a graph of g. (i) lim g(x) (ii) lim g(x) (iii) lim g(x) x0- (iv) lim g

> The signum (or sign) function, denoted by sgn, is defined by a. Sketch the graph of this function. b. Find each of the following limits or explain why it does not exist. -1 if x<0 0 if x= 0 1 if x>0 sgn x = (i) lim sgn x x0+ (ii) lim sgn x X0- (i

> Find the limit, if it exists. If the limit does not exist, explain why. lim X0+ |x|/

> Sketch the graph of the function and use it to determine the values of a for which limx &acirc;&#134;&#146; a f(x) exists. 1 +x if x<-1 if -1 <x<1 2 - x if x>1 f(x) = {x

> Find the limit, if it exists. If the limit does not exist, explain why. 1 lim |x|) X0-

> Find the limit, if it exists. If the limit does not exist, explain why. 2 - |x| lim x→-2 2 + x

> Find the limit, if it exists. If the limit does not exist, explain why. 2x lim x-0.5- |2x – x²|

> Find the limit, if it exists. If the limit does not exist, explain why. 2x + 12 lim x-6 |x + 6|

> Find the limit, if it exists. If the limit does not exist, explain why. lim (2x + |x – 3|)

> Prove that lim r esin(/3) = 0.

> Prove that lim x* cos 2 0.

> If 2x &acirc;&#137;&curren; g(x) &acirc;&#137;&curren; x4 - x2 + 2 for all x, evaluate lim g(x).

> If 4x - 9 &acirc;&#137;&curren; f(x) &acirc;&#137;&curren; x2 - 4x + 7 for x &acirc;&#137;&yen; 0, find lim f(x). X4

> Use the Squeeze Theorem to show that Illustrate by graphing the functions f, g, and h (in the notation of the Squeeze Theorem) on the same screen. lim Vx3 + x² sin TT %3D

> A patient receives a 150-mg injection of a drug every 4&Acirc;&nbsp;hours. The graph shows the amount f(t) of the drug in the blood stream after t hours. Find and explain the significance of these one-sided limits. lim f(1) lim f(1) +12 and +12 f(

> A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. T

> A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes. a. If P is the point (15, 250) on the graph of V, find

> Find an equation of the parabola with focus (2, 1) and directrix x = -4.

> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan-1x, and ln(1 + x)

> Determine whether the sequence converges or diverges. If it converges, find the limit. an Vn + 2

> Use the Ratio Test to determine whether the series is convergent or divergent. 3" E (-1)"-1 2"n n-1

> Evaluate the integral. 1- tan?x sec?x 2,

> Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan-1x, and ln(1 + x)

> Find the foci and vertices and sketch the graph. бу? + х — 36у + 55 — 0

> (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson&acirc;&#128;&#153;s Rule with n = 10 to approximate the given integral. Round your answers to six decimal places. dx J2 In x a>

> The force due to gravity on an object with mass m at a height h above the surface of the earth is where R is the radius of the earth and t is the acceleration due to gravity for an object on the surface of the earth. (a) Express F as a series in powers o

> Use series to evaluate the following limit. sin x – x lim .3

> (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Graph f and Tn on a common screen. (c) Use Taylor&acirc;&#128;&#153;s Inequality to estimate the accuracy of the approximation / when x lies in the given interval. (d) Check your

> (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Graph f and Tn on a common screen. (c) Use Taylor&acirc;&#128;&#153;s Inequality to estimate the accuracy of the approximation / when x lies in the given interval. (d) Check your

1.99

See Answer