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


Is there a number a such that

Exists? If so, find the value of a and the value of the limit.

The figure shows a fixed circle C1 with equation (x – 1)2 + y2 = 1 and a shrinking circle C2 with radius and center the origin. P is the point (0, r), Q is the upper point of intersection of the two circles, and R is the point of intersection of the line PQ and the -axis. What happens to R as C2 shrinks, that is, as r → 0+?




How close to 2 do we have to take so that 5x + 3 is within a distance of
(a) 0.1 and
(b) 0.01 from 13?
How close to 5 do we have to take so that 6x – 1 is within a distance of
(a) 0.01,
(b) 0.001, and
(c) 0.0001 from 29?
Use the given graph of f(x) = 1/x to find a number such that


Use the given graph of f to find a number δ such that



Use the given graph of f(x) = √x to find a number δ such that |√x – 2|



Use the given graph of f(x) = x2 to find a number δ such that |x2 – 1|



Use a graph to find a number such that |√4x + 1 3| < 0.5 whenever |x – 2| <δ
Use a graph to find a number such that |sin x – ½| < 0.1 whenever |x – π/6| < δ
For the limit lim x→1 (4 + x – 3x3) = 2 illustrate Definition 2 by finding values of that correspond to ε = 1 and ε = 0.1.
For the limit lim x→0 ex – 1/x = 1 illustrate Definition 2 by finding values of that correspond to ε = 0.5 and ε = 0.1.
Use a graph to find a number δ such that x/(x2 + 1) (x – 1)2 > 100 whenever 0 < |x – 1| <δ
For the limit




Illustrate Definition 6 by finding values of δ that correspond to
(a) M = 100 and
(b) M = 1000.
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 the machinist control the radius?
(c) In terms of the ε, δ definition of lim x→ a f(x) = L, what is x? What is f(x)? What is a? What is L? What value of is given? What is the corresponding value of δ?
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. Suppose the relationship is given by T(w) = 0.1w2 + 2.155w + 20 where T is the temperature in degrees Celsius and w is the power input in watts. (a) How much power is needed to maintain the temperature at 200oC? (b) If the temperature is allowed to vary from 200oC by up to ±1oC, what range of wattage is allowed for the input power? (c) In terms of the ε, δ definition of lim x→ a f(x) = L, what is x? What is f(x)? What is a? What is L? What value of is given? What is the corresponding value of δ?
Prove the statement using the ε, δ definition of limit and illustrate with a diagram like Figure 9.




Prove the statement using the ε, δ definition of limit.



Verify that another possible choice of for showing that lim x →3 x2 = 9 in Example 4 is δ = min {2, ε/8}.
Verify, by a geometric argument, that the largest possible choice of δ for showing that lim x→3x2 = 9 is δ = √ 9 is ε – 3.
(a) For the limit lim x→1(x3 + x + 1) = 3, use a graph to find a value of 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 that works for any given ε > 0.
(c) Put ε = 0.4 in your answer to part (b) and compare with your answer to part (a).
Prove that lim x→2 1/x = ½.
Prove that lim x→ a √x = √a if a > 0.
If H is the Heaviside function defined in Example 6 in Section 2.2, prove, using Definition 2, that lim t→ H(t) does not exist.
If the function f is defined by




Prove that limx→0 f (x) does not exist.
By comparing Definitions 2, 3, and 4, prove Theorem 1 in Section 2.3.
How close to – 3 do we have to take so that


Prove, using Definition 6, that lim x →-3 1/(x + 3)4 =∞.
Prove that lim x →0+ in x = - ∞.
Suppose that lim x →a f(x) = ∞ and lim x →a f(x) = c, where is a real number.
Prove each statement.
(a) lim x →a [f(x) + g(x)] = ∞
(b) lim x →a [f(x)g(x)] = ∞ if c > 0.
(c) lim x →a [f(x)g(x)] = – ∞ if c < 0.
Write an equation that expresses the fact that a function f is continuous at the number 4.
If f is continuous on (– ∞, ∞), what can you say about its graph?
(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.




From the graph of g, state the intervals on which is continuous


Sketch the graph of a function that is continuous everywhere except at x = 3 and is continuous from the left at 3.
Sketch the graph of a function that has a jump discontinuity at x = 2 and a removable discontinuity at x = 4, but is continuous elsewhere.
A parking lot charges $3 for the first hour (or part of an hour) and $2 for each succeeding hour (or part), up to a daily maximum of $10.
(a) Sketch a graph of the cost of parking at this lot as a function of the time parked there.
(b) Discuss the discontinuities of this function and their significance to someone who parks in the lot.
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 a function of the distance due west from New York City
(d) The cost of a taxi ride as a function of the distance traveled
(e) The current in the circuit for the lights in a room as a function of time
If f and are continuous functions with f(3) = 5 and lim x→3 [2f(x)] – g(x)] = 4, find g(3).
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number.


Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.


Explain why the function is discontinuous at the given number a. Sketch the graph of the function.


Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.


Locate the discontinuities of the function and illustrate by graphing.


Use continuity to evaluate the limit.


Show that f is continuous on (– ∞, ∞).



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 of f.


The gravitational force exerted by 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?

For what value of the constant is the function continuous on (– ∞, ∞)?




Find the constant that makes continuous on (– ∞, ∞)?



Which of the following functions f has a removable discontinuity at a? If the discontinuity is removable, find a function g that agrees with f for x ≠ a and is continuous on R.




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 might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesn’t satisfy the hypothesis).
If f(x) = x3 – x2 + x, show that there is a number such that f(c) = 10.
Use the Intermediate Value Theorem to prove that there is a positive number such that c2 = 2. (This proves the existence of the number √2.
Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.


(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.


(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.


Prove that f is continuous at if and only if


To prove that sine is continuous, we need to show that lim x→ a sin x – sin a for every real number . By Exercise 55 an equivalent statement is that




Use (6) to show that this is true.
Prove that cosine is a continuous function.
(a) Prove Theorem 4, part 3.
(b) Prove Theorem 4, part 5
For what values of is continuous?


Is there a number that is exactly 1 more than its cube?
(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 Tibetan monk leaves the monastery at 7:00 A.M. and takes his usual path to the top of the mountain, arriving at 7:00 P.M. The following morning, he starts at 7:00 A.M. at the top and takes the same path back, arriving at the monastery at 7:00 P.M. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days.
For what values of is continuous?


Explain in your own words the meaning of each of the following.


For the function whose graph is given, state the following.
(a) lim x →2 f(x) (b) lim x →-1- f(x) (c) lim x →-1+ f(x)
(d) lim x →∞ f(x) (e) lim x →-∞ f(x)
(f) The equations of the asymptotes




For the function t whose graph is given, state the following.
(a) lim x →∞ g(x) (b) lim x →-∞ g(x)
(c) lim x →3 g(x) (d) lim x →-0 g(x)
(c) lim x →-2+ g(x) (d) The equations of the asymptotes




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


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 to support your guess

(a) Use a graph of




to estimate the value of lim x→∞ f(x) correct to two decimal places.
(b) Use a table of values of f(x) to estimate the limit to four decimal places.
Evaluate the limit and justify each step by indicating the appropriate properties of limits.


Find 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.
(a) Use a graph of




to estimate the value of lim x→∞ 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.
Find the horizontal and vertical asymptotes of each curve. Check your work by graphing the curve and estimating the asymptotes.


Find a formula for a function that satisfies the following conditions.


Find a formula for a function that has vertical asymptotes x = 1 and x = 3 and horizontal asymptote y = 1.
Find the limits as x → ∞ and as. Use this x → -∞ information, together with intercepts, to give a rough sketch of the graph as in Example 11.




(a) Use the Squeeze Theorem to evaluate lim x →∞ sin x / x.
(b) Graph f(x) = (sin x)/x. How many times does the graph cross the asymptote?
By the end behavior of a function we mean the behavior of its values as x →∞ and as x →∞.
(a) Describe and compare the end behavior of the functions by graphing both functions in the viewing




rectangles [ –2, 2] by [ –2, 2] and [ –10, 10] by . [–10,000, 10,000]
(b) Two functions are said to have the same end behavior if their ratio approaches 1 as x → ∞. Show that P and Q have the same end behavior.
Let P and Q be polynomials. Find

If the degree of P is
(a) Less than the degree of Q and
(b) Greater than the degree of Q.

Make a rough sketch of the curve y = xn (n an integer) for the following five cases:
(i) n = 0 (ii) n > 0, n odd
(iii) n > 0, n even (iv) n (v) n > 0, n even
Then use these sketches to find the following limits.


Find lim x→∞ f(x) if




For all x > 5
(a) A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. Show that the concentration of salt after t minutes (in grams per liter) is



(b) What happens to the concentration as t →∞?
(a) Can the graph of y = f(x) intersects a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs.
(b) How many horizontal asymptotes can the graph of y = f(x) have? Sketch graphs to illustrate the possibilities.
In Chapter 9 we will be able to show, under certain assumptions, that the velocity v(t) of a falling raindrop at time t is




where t is the acceleration due to gravity and is the terminal velocity of the raindrop.
(a) Find lim t→∞ v(t).
(b) Graph v(t) if v* = 1 m/s and g = 9.8m/s2. How long does it take for the velocity of the raindrop to reach 99% of its terminal velocity?
(a) By graphing y = e –x/10 and y = 0.1 on a common screen, discover how large you need to make x so that e–x/10 < 0 .1.
(b) Can you solve part (a) without using a graphing device?
Use a graph to find a number N such that


For the limit illustrate




Definition 7 by finding values N of that correspond to ε = 0.5 and ε = 0.1
For the limit illustrate




Definition 8 by finding values of N that correspond to ε = 0.5 and ε = 0.1.
For the limit illustrate




Definition 9 by finding a value of N that corresponds to M = 100.
(a) How large do we have to take so that 1/x2 (b) Taking r = 2 in Theorem 5, we have the statement

Prove this directly using Definition 7.

(a) How large do we have to take so that 1/√x (b) Taking r = ½ in Theorem 5, we have the statement




Prove this directly using Definition 7.
Use Definition 8 to prove that lim x→∞ 1/x = 0.
Prove, using Definition 9, that lim x→∞ x3 = ∞
Use Definition 9 to prove that lim x→∞ ex = ∞
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