1.99
See Answer

** >** Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = (3 - x)(1 + x)2(1 - x)4

** >** Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = x3(x + 2)2(x - 1)

** >** Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = x4 - x6

** >** Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = 2x3 - x4

** >** A function f is a ratio of quadratic functions and has a vertical asymptote x = 4 and just one x-intercept, x = 1. It is known that f has a removable discontinuity at x = -1 and limx â†’-1 f(x) = 2. Evaluate a. f(0)
(b) lim f(x)

** >** a. Estimate the value of by graphing the function f(x) = (sin Ï€x)/(sin Ï€x). State your answer correct to two decimal places. b. Check your answer in part (a) by evaluating f(x) for values of x that approach 0.
sin x lim X0 si

** >** The point Ps2, 21d lies on the curve y = 1/(1 – x). a. If Q is the point (x, 1/(1 - x)), use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x: i. 1.5 ii. 1.9 iii. 1.99 iv. 1.999 v

** >** Find a formula for a function that has vertical asymptotes x = 1 and x = 3 and horizontal asymptote y = 1.

** >** Find a formula for a function f that satisfies the following conditions:
lim f(x) = 0, lim f(x) = -, f(2) = 0, %3D lim f(x) = ∞, lim f(x) = -00

** >** 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 Then use these sketches to find the following limits.
(а) lim x^ X0+ (b) lim x" X0- (c) lim x" (d)

** >** 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.
P(x) lim Q(x)

** >** a. Graph the function f(x) = 2x2 + 1 /3x - 5 How many horizontal and vertical asymptotes do you observe? Use the graph to estimate the values of the limits b. By calculating values of f(x), give numerical estimates of the limits in part (a). c. Cal

** >** Estimate the horizontal asymptote of the function f(x) = 3x3 + 500x2/x3 + 500x2 + 100x + 2000 by graphing f for -10 ≤ x ≤ 10. Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?

** >** 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 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

** >** 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

** >** 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 (-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