1.99
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Prove the statement using the Îµ, Î´ 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 definition of continuity and the properties of limits to show that the function is continuous on the given interval. f(x) = x + x − 4 , (4, ∞)

** >** Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. f(x) = 3x4 - 5x + 3 x2 + 4 , a = 2

** >** Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. p(v) = 2 3v2 + 1 , a = 1

** >** Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. g(t) = t2 + 5t/2t + 1, a = 2

** >** Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. f(x) = (x + 2x3)4, a = -1

** >** Explain why each function is continuous or discontinuous. a. The temperature at a specific location as a function of time b. The temperature at a specific time as a function of the distance due west from New York City c. The altitude above sea level as

** >** Sketch the graph of an example of a function f that satisfies all of the given conditions.
lim f(x) = 2, lim f(x) = 0, lim f(x)= 3, lim f(x) = 0, f(0) = 2, f(4) = 1

** >** The toll T charged for driving on a certain stretch of a toll road is $5 except during rush hours (between 7 am and 10 am and between 4 pm and 7 pm) when the toll is $7. a. Sketch a graph of T as a function of the time t, measured in hours past midnigh

** >** 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 Î´ such that if |x - 1|
1.5- y =x? 1 0.5 ? ?

** >** Use the given graph of f(x) = âˆšx to find a number such that if |x - 4| Î´ then | x âˆ’ 2 |
yA ソ=Vx 2.4 2 1.6 ? 4 ?

** >** Use the given graph off to find a number Î´ 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 Î´ 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 limxâ†’a f(x) = âˆž and limxâ†’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 limxâ†’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 Îµ, Î´ definition of a limit.
lim x3 = 8

** >** Prove the statement using the Îµ, Î´ definition of a limit.
lim (x2 – 1) = 3 X-2

** >** Prove the statement using the Îµ, Î´ definition of a limit.
lim (x? + 2x – 7) = 1 %3D

** >** Prove the statement using the Îµ, Î´ definition of a limit.
lim (x? – 4x + 5) = 1

** >** Prove the statement using the Îµ, Î´ definition of a limit.
lim 16 + x = 0 X→-6+

** >** Prove the statement using the Îµ, Î´ definition of a limit.
lim |x| = 0

** >** Prove the statement using the Îµ, Î´ definition of a limit.
lim x = 0 .3

** >** Prove the statement using the Îµ, Î´ 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 Îµ, Î´ definition of a limit.
lim c = c

** >** Prove the statement using the Îµ, Î´ definition of a limit.
lim x = a

** >** Prove the statement using the Îµ, Î´ definition of a limit.
9 – 4x2 lim X→-1.5 3 + 2x

** >** Prove the statement using the Îµ, Î´ definition of a limit.
x2 lim — 2х — 8 х — 4

** >** Prove the statement using the Îµ, Î´ definition of a limit.
(3 – x) = -5 X10

** >** Prove the statement using the Îµ, Î´ definition of a limit.
2 + 4x = 2 3 lim

** >** Prove the statement using the Îµ, Î´ 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 Îµ, Î´ 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 Îµ, Î´ 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

** >** 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 Îµ = 0.5 and Îµ = 0.1.
e 2x lim 1 = 2

** >** For the limit illustrate Definition 2 by finding values of that correspond to Îµ= 0.2 and Îµ = 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)

** >** 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.

** >** 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.

** >** If prove that limxâ†’0 f(x) = 0.
|x² if x is rational if x is irrational f(x) =

** >** If / f(x) x2 = 5, find the following limits.
lim

** >** If / f(x) – 8/x - 1 = 10, find / f(x).

** >** If r is a rational function, use Exercise 57 to show that limx→a r(x) = r(a) for every number a in the domain of r.

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

** >** In the theory of relativity, the Lorentz contraction formula expresses the length L of an object as a function of its velocity v with respect to an observer, where L0 is the length of the object at rest and c is the speed of light. Find limvâ

** >** Sketch the graph of the function and use it to determine the values of a for which limx â†’ a f(x) exists.
|1 + sin x if x <0 S(x) cos x if 0<x<T sin x if x>T

** >** If f(x) = [[x]] + [[2x]], show that limx→2 f(x) exists but is not equal to f(2).

** >** Let f(x) = [[cos x]], -Ï€ â‰¤ x â‰¤ Ï€ . a. Sketch the graph off. b. Evaluate each limit, if it exists. c. For what values of a does limx â†’ 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 â†’ 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)

** >** Let a. Find limxâ†’1- f(x) and limxâ†’1+ f(x). b. Does limxâ†’1 f(x) exist? c. Sketch the graph of f.
(x² + 1 if x<1 l(x – 2)? if x> 1 f(x) =

** >** Let g(x) = x2 + x â€“ 6/|x - 2| a. Find b. Does limx â†’ 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 â†’ 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 â†’ 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 â‰¤ g(x) â‰¤ x4 - x2 + 2 for all x, evaluate
lim g(x).

** >** If 4x - 9 â‰¤ f(x) â‰¤ x2 - 4x + 7 for x â‰¥ 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 4Â 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