- •Functions
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- •Logarithmic, Exponential and Hyperbolic Functions
- •Limits and Continuity
- •Introductory Examples
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- •The Chain Rule
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- •Mathematical Applications
- •Antidifferentiation
- •Linear Second Order Homogeneous Differential Equations
- •Linear Non-Homogeneous Second Order Differential Equations
- •Area Approximation
- •Integration by Substitution
- •Integration by Parts
- •Logarithmic, Exponential and Hyperbolic Functions
- •The Riemann Integral
- •Volumes of Revolution
- •Arc Length and Surface Area
- •Techniques of Integration
- •Integration by formulae
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- •Integration by Parts
- •Trigonometric Integrals
- •Trigonometric Substitutions
- •Integration by Partial Fractions
- •Fractional Power Substitutions
- •Numerical Integration
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- •Discontinuities at End Points
- •Improper Integrals
- •Sequences
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- •Areas in Polar Coordinates
- •Parametric Equations
7.2. DISCONTINUITIES AT END POINTS |
299 |
Then F (b) −F (a) represents the percentage of normally distributed data that lies between a and b. This percentage is given by
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Furthermore, |
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/2dx. |
2π |
Proof. The proof of this theorem is omitted.
Exercises 7.1 None available.
7.2Discontinuities at End Points
Definition 7.2.1 (i) Suppose that f is continuous on [a, b) and
lim f(x) = +∞ or − ∞.
x→b−
Then, we define
Z b Z x
f(x)dx = lim f(x)dx.
ax→b− a
If the limit exists, we say that the improper integral converges; otherwise we say that it diverges.
(ii) Suppose that f is continuous on (a, b] and
x→a+ |
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or |
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lim f(x) = + |
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Then we define, |
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Z b f(x)dx. |
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Z b f(x)dx = |
lim |
ax→a+ x
If the limit exists, we say that the improper integral converges; otherwise we say that it diverges.
Exercises 7.2
1.Suppose that f is continuous on (−∞, ∞) and g0(x) = f(x). Then define each of the following improper integrals:
300CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS
Z+∞
(a)f(x)dx
a
Zb
(b)f(x)dx
−∞
Z+∞
(c)f(x)dx
−∞
2.Suppose that f is continuous on the open interval (a, b) and g0(x) = f(x) on (a, b). Define each of the following improper integrals if f is not continuous at a or b:
Zx
(a)f(x)dx, a ≤ x < b
a
Zb
(b)f(x)dx, a < x ≤ b
x
Zb
(c)f(x)dx
a
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Prove that Z0 |
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Prove that Z0 |
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4. |
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+∞ |
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5. |
Prove that Z−∞ |
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6. |
Prove that Z1 |
∞ 1 |
dx = |
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7. |
Show that Z−∞ |
e−x |
dx = 2 Z0 |
e−x |
dx. Use the comparison between |
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e−x and e−x |
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8. |
Prove that Z0 |
1 dx |
converges if and only if p < 1. |
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7.2. |
DISCONTINUITIES AT END POINTS |
301 |
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9. |
Evaluate Z0 |
+∞ e−x sin(2x)dx. |
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10. |
Evaluate Z0 |
+∞ e−4x cos(3x)dx. |
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11. |
Evaluate Z0 |
+∞ x2e−xdx. |
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12. |
Evaluate Z0 |
+∞ xe−xdx. |
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13. |
Prove that Z0 |
∞ sin(2x)dx diverges. |
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14. |
Prove that Z0 |
∞ cos(3x)dx diverges. |
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15.Compute the volume of the solid generated when the area between the graph of y = e−x2 and the x-axis is rotated about the y-axis.
16.Compute the volume of the solid generated when the area between the graph of y = e−x, 0 ≤ x < ∞ and the x-axis is rotated
(a)about the x-axis
(b)about the y-axis.
17.Let A represent the area bounded by the graph y = x1, 1 ≤ x < ∞ and the x-axis. Let V denote the volume generated when the area A is rotated about the x-axis.
(a)show that A is +∞
(b)show that V = π
(c)show that the surface area of V is +∞.
(d)Is it possible to fill the volume V with paint and not be able to paint its surface? Explain.
18.Let A represent the area bounded by the graph of y = e−2x, 0 ≤ x < ∞, and y = 0.
302CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS
(a)Compute the area of A.
(b)Compute the volume generated when A is rotated about the x-axis.
(c)Compute the volume generated when A is rotated about the y-axis.
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Assume that Z0 |
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20. |
It is known that Z−∞ |
e−x |
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(a) Compute Z0 |
+∞ e−x2 dx. |
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(b) Compute Z0 |
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(c) Compute Z0 |
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Definition 7.2.2 Suppose that f(t) is continuous on [0, ∞) and there exist some constants a > 0, M > 0 and T > 0 such that |f(t)| < Meat for all t ≥ T . Then we define the Laplace transform of f(t), denoted L{f(t)}, by
Z ∞
L{f(t)} = e−stf(t)dt
0
for all s ≥ s0. In problems 21–34, compute L{f(t)} for the given f(t).
21. f(t) = |
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if t ≥ 0 |
22. f(t) = t |
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if t < 0 |
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23. f(t) = t2
25. f(t) = tn, n = 1, 2, 3, · · ·
27. f(t) = tebt
7.2. DISCONTINUITIES AT END POINTS |
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29. |
f(t) = |
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30. |
f(t) = |
aeat − bebt |
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a − b |
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31. |
f(t) = |
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32. |
f(t) = cos(bt) |
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33. |
f(t) = |
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34. |
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Definition 7.2.3 For x > 0, we define the Gamma function (x) by |
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In problems 35–40 assume that (x) exists for x > 0 and Z0 |
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35. |
Show that (1/2) = √ |
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36. |
Show that (1) = 1 |
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37. |
Prove that (x + 1) = x (x) |
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38. |
Show that |
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39. |
Show that |
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40. |
Show that (n + 1) = n! |
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In problems 41–60, evaluate the given improper integrals. |
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41. |
Z0 |
2xe−x |
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304CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS
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49. |
Z−∞ |
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50. |
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∞ e− |
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59. |
Z0 |
∞ |
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60. |
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+∞ x2e−x3 dx |
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7.3
Theorem 7.3.1 (Cauchy Mean Value Theorem) Suppose that two functions f and g are continuous on the closed interval [a, b], di erentiable on the open interval (a, b) and g0(x) 6= 0 on (a, b). Then there exists at least one number c such that a < c < b and
f0(c) f(b) − f(a) g0(c) = g(b) − g(a) .
Proof. See the proof of Theorem 4.1.6.
Theorem 7.3.2 Suppose that f and g are continuous and di erentiable on an open interval (a, b) and a < c < b. If f(c) = g(c) = 0, g0(x) 6= 0 on (a, b) and
lim f0(x) = L
x→c g0(x)
then
lim f(x) = L.
x→c g(x)
7.3. 305
Proof. See the proof of Theorem 4.1.7.
Theorem 7.3.3 (L’Hˆopital’s Rule) Let lim represent one of the limits
lim, |
lim , |
lim , |
lim , or |
lim . |
x→c |
x→c+ |
x→c− |
x→+∞ |
x→−∞ |
Suppose that f and g are continuous and di erentiable on an open interval (a, b) except at an interior point c, a < c < b. Suppose further that g0(x) 6= 0 on (a, b), lim f(x) = lim g(x) = 0 or lim f(x) = lim g(x) = +∞ or −∞. If
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then |
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g0(x) |
Proof. The proof of this theorem is omitted.
Definition 7.3.1 (Extended Arithmetic) For the sake of convenience in dealing with indeterminate forms, we define the following arithmetic operations with real numbers, +∞ and −∞. Let c be a real number and c > 0. Then we define
+ ∞ + ∞ = +∞, −∞ − ∞ = −∞, c(+∞) = +∞, c(−∞) = −∞
( c)(+ ) = |
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( c)( |
) = + , |
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= 0, |
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= 0, |
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Definition 7.3.2 The following operations are indeterminate: |
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+∞ −∞ −∞ |
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Remark 23 The L’Hˆopital’s Rule can be applied directly to the 00 and ±∞±∞ forms. The forms ∞ − ∞ and 0 · ∞ can be changed to the by using arithmetic operations. For the 00 and 1∞ forms we use the following procedure:
lim(f(x))g(x) = lim eg(x) ln(f(x)) = elim
It is best to study a lot of examples and work problems.
306CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS
Exercises 7.3
1.Prove the Theorem of the Mean: Suppose that a function f is continuous on a closed and bounded interval [a, b] and f0 exists on the open interval (a, b). Then there exists at least one number c such that a < c < b and
(1) |
f(b) |
− f(a) |
= f0(c) |
(2) f(b) = f(a) + f0(c)(b |
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a). |
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2. Prove the Generalized Theorem of the Mean: Suppose that f and g are continuous on a closed and bounded interval [a, b] and f0 and g0 exist on the open interval (a, b) and g0(x) 6= 0 for any x in (a, b). Then there exists some c such that a < c < b and
f(b) − f(a) |
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f0(c) |
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g(b) |
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3.Prove the following theorem known as l’Hˆopital’s Rule: Suppose that f and g are di erentiable functions, except possibly at a, such that
lim f(x) = 0, |
lim g(x) = 0, |
and |
lim |
f(x) |
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Then |
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f0(x) |
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lim |
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4.Prove the following theorem known as an alternate form of l’Hˆopital’s Rule: Suppose that f and g are di erentiable functions, except possibly at a, such that
lim f(x) = |
∞ |
, |
lim g(x) = |
∞ |
, and lim |
f0(x) |
= L. |
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7.3. |
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307 |
5. Prove that if f0 and g0 exist and |
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lim |
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lim g(x) = 0, |
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lim |
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x→+∞ |
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then |
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f(x) |
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6. Prove that if f0 and g0 exist and |
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lim f(x) = 0, |
lim g(0) = 0, |
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lim |
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x→−∞ g0(x) |
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then |
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f(x) |
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7. Prove that if f0 and g0 exist and |
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lim |
f(x) = |
∞ |
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lim g(x) = |
∞ |
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and |
lim |
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f0(x) |
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8. Prove that if f0 and g0 |
exist and |
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x lim |
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x lim |
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9.Suppose that f0 and f00 exist in an open interval (a, b) containing c. Then prove that
lim |
f(c + h) − 2f(c) + f(c − h) |
= f00(c). |
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h2 |
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h→0 |
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308CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS
10. Suppose that f0 is continuous in an open interval (a, b) containing c. Then prove that
lim f(c + h) − f(c − h) = f0(c).
h→0 2h
11. Suppose that f(x) and g(x) are two polynomials such that
f(x) = a0xn + a1xn−1 + · · · + an−1x + an, a0 6= 0, g(x) = b0xm + b1xm−1 + · · · + bm−1x + bm, b0 6= 0.
Then prove that |
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lim f(x) = |
if m < n |
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x→+∞ g(x) |
if m = n |
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a0/b0 |
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12.Suppose that f and g are di erentiable functions, except possibly at c, and
lim f(x) = 0, |
lim |
g(x) = 0 and lim g(x) ln(f(x)) = L. |
x→c |
x→c |
x→c |
Then prove that |
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lim (f(x))g(x) = eL. |
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x→c |
13.Suppose that f and g are di erentiable functions, except possibly at c, and
lim f(x) = + |
∞ |
, |
lim g(x) = 0 and |
lim g(x) ln(f(x)) = L. |
x→c |
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x→c |
x→c |
Then prove that
lim (f(x))g(x) = eL.
x→c
14.Suppose that f and g are di erentiable functions, except possibly at c, and
lim f(x) = 1, |
lim g(x) = + |
∞ |
and |
lim g(x) ln(f(x)) = L. |
x→c |
x→c |
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x→c |
Then prove that
lim (f(x))g(x) = eL.
x→c
7.3. |
309 |
15.Suppose that f and g are di erentiable functions, except possibly at c, and
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lim f(x) = 0, |
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lim g(x) = + |
and |
lim |
f(x) |
= |
L. |
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Then prove that |
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16. |
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1 |
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17. |
Prove that lim (1 |
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Prove that |
lim |
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19. |
Prove that |
lim |
sin x − x |
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20. |
Prove that lim |
π |
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In problems 21–50 evaluate each of the limits.
21. |
lim |
sin(x2) |
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22. |
lim |
1 − cos x2 |
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x2 |
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23. |
lim |
sin(ax) |
24. |
lim |
tan(mx) |
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25. |
lim |
e3x − 1 |
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26. |
lim (1 + 2x)3/x |
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27. |
lim |
ln(x + h) − ln(x) |
28. |
lim |
ex+h − ex |
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29. |
lim (1 + mx)n/x |
30. |
lim |
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310CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS
31. lim (1 + sin mx)n/x
x→0
33.lim (x)sin x
x→0+
35.lim tan(2x) ln(x)
x→0+
37. |
lim (x + ex)2/x |
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39. |
lim (1 + sin mx)n/x |
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41. |
lim |
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1)2/ ln x |
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43. |
lim |
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lim |
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47. |
lim |
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49. |
h→0 |
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, b > |
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51. |
lim |
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53. |
lim |
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55. |
x→+∞ |
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ex + 1 |
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ex |
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lim |
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ex ln |
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32.lim (sin x)x
x→0+
34. lim
x4 − 2x3 + 10
x→∞ 3x4 + 2x3 − 7x + 1
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lim |
x sin |
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38. |
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3 + 2x |
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x→∞ 4 + 2x |
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lim |
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40. |
lim (x)sin(3x) |
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x→0+ |
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lim |
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42. |
x→0 |
x2 − |
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44. |
lim |
ln x |
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x |
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x→+∞ |
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2 |
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lim |
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46. |
x→0+ |
x − ln x |
48.lim x(b1/x − 1), b > 0, b 6= 1
x→+∞
50. lim |
logb(x + h) − logb x |
, b > 0, b = 1 |
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h |
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h |
6 |
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52. |
lim |
x ln |
x + 1 |
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x − 1 |
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x→+∞ |
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54. |
lim |
2x − 3x6 + x7 |
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(1 − x)3 |
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x→1 |
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56. |
lim |
tan x − sin x |
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x→0 |
x3 |
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7.3.
57. lim
x3 sin 2x
x→0 (1 − cos x)2
59. |
lim |
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ln |
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1 + x |
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1 − x |
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61. |
lim |
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x sin x |
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x→0 |
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63. |
lim |
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(ln x)n |
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· · · |
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x→+∞ |
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65. |
lim |
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ln x |
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(1 + x3)1/2 |
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67. |
lim |
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3−x)−2x |
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69. |
lim |
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(e−x + e−2x)1/x |
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71. |
x→0+ |
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ln |
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lim |
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73. |
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1 |
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3x+ln x |
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x→+∞ |
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2x |
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lim |
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1 + |
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75. |
x→+∞ |
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− |
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√x2 + b2 |
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lim |
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77. |
x→2 |
x − 2 − x2 + x − 6 |
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lim |
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1 |
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5 |
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1 |
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lim |
cot x |
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79. |
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− x |
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x→0 |
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311
58. |
lim |
5x − 3x |
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x→0 |
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x2 |
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60. |
lim |
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arctan x − x |
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x→0 |
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x3 |
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62. |
lim |
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ln(1 + xe2x) |
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x2 |
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x→+∞ |
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64. |
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1 |
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ln |
x + e2x |
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x→+∞ √x |
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x |
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lim |
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ln(tan 3x) |
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66. |
lim |
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ln(tan 4x) |
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x→0+ |
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68. |
x→0 |
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sin x |
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1/x2 |
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x |
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70. |
lim |
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x2 |
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x→+∞ |
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cos x |
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lim |
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3 |
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72. |
x→+∞ |
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1 |
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x2 |
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2x |
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lim |
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lim |
1 |
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1 |
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74. |
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x − sin 2x |
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x→0 |
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lim |
1 |
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1 |
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76. |
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x sin x − x2 |
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x→0 |
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lim |
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78. |
x→0+ |
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x − ln |
x |
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lim |
1 |
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1 |
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80. |
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x2 − tan2 x |
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x→0 |
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312CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS
81. |
x→0 |
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e−x |
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1 |
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x |
− ex − 1 |
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lim |
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x2 sin |
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1 |
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→ |
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sin |
x |
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83. |
lim |
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x 0 |
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x |
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85. |
lim |
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e − (1 + x)1/x |
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x→0 |
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x |
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lim |
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87. |
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x2 − x ln x |
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x→0+ |
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89. |
lim |
(ln(1 + ex) |
− |
x) |
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x→+∞ |
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82. |
lim |
x − sin x |
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x→∞ |
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x |
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x→∞ |
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x |
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84. |
lim x sin |
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86. |
lim |
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ln(ln x) |
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ln(x − ln x) |
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x→+∞ |
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88. |
x→+∞ |
1 |
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x |
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ln t |
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x Z1 |
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1 + t |
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lim |
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90. |
x→+∞ x2 |
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sin |
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Z0 |
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lim |
1 |
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2 x dx |
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91.Suppose that f is defined and di erentiable in an open interval (a, b). Suppose that a < c < b and f00(c) exists. Prove that
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f00(c) = lim |
f(x) − f(c) − (x − c)f0(c) |
. |
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x→c |
((x − c)2/2!) |
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92. Suppose that f is defined and f0, f00, |
· · · |
(n) |
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, f(n−1) exist in an open interval |
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(a, b). Also, suppose that a < c < b and f (c) exists |
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(a) Prove that |
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f(x) − f(c) − (x − c)f0(c) − · · · − |
(x−c)n−1 |
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n |
1 |
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(n) |
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(n−1)! |
f |
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− |
(c) |
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f |
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(c) = lim |
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(x−c) |
n |
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. |
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x |
→ |
c |
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n! |
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(b)Show that there is a function En(x) defined on (a, b), except possibly at c, such that
f(x) = f(c) + (x − c)f0(c) + · · · + (x − c)n−1 f(n−1)(x) (n − 1)!
+ (x − c)n f(n)(c) + En(x) (x − c)nEn(x) n! n!
7.3. |
313 |
and lim En(x) = 0. Find E2(x) if c = 0 and |
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n→c |
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f(x) = (x4 sin x1 |
, x 6= 0 |
0, x = 0
(c)If f0(c) = · · · = f(n−1)(c) = 0, n is even, and f has a relative minimum at x = c, then show that f(n)(c) ≥ 0. What can be said if f has a relative maximum at c? What are the su cient conditions for a relative maximum or minimum at c when f0(c) = · · · = f(n−1)(c) = 0?
What can be said if n is odd and f0(c) = |
· · · |
= f(n−1) |
(c) = 0 but |
f(n)(c) 6= 0. |
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93.Suppose that f and g are defined, have derivatives of order 1, 2, · · · , n−1 in an open interval (a, b), a < c < b, f(n)(c) and g(n)(c) exist and g(n)(c) 6= 0. Prove that if f and g, as well as their first n − 1 derivatives are 0, then
Evaluate the following limits:
94. |
x→0 |
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x2 sin 1 |
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x |
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lim |
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x |
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96. lim x(1−1x )
x→1
98. lim
xx − x
x→1+ 1 − x + ln x
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f(x) |
f(n)(c) |
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lim |
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= |
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. |
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x→c |
g(x) |
g(n)(c) |
→ |
cos |
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π cos x |
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95. lim |
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2 |
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sin |
2 |
x |
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x 0 |
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97.lim x(ln(x))n, n = 1, 2, 3, · · ·
x→0+
99. lim
x3/2 ln x
x→+∞ (1 + x4)1/2
100. |
x→+∞ |
xn |
ln |
1 + ex |
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= 1 2 · · · |
|||
ex |
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lim |
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, n |
, , |
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x |
1R |
x e−t2 dx |
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x→0 |
− e−x2 |
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101. |
lim |
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0 |
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