ลิมิตตรีโกณ 56.pdf
TRANSCRIPT
-
,
lim lim sin
lim cos2.=........................
krupornchaikrupornchaikrupornchaikrupornchai
y y = sin = cos
x
-2
-1
1
2
-
2 2
3
2 - 0
2
x
-2
-1
1
2
-2
23
2 - 0
2
x
-2
-1
1
2
-
2 2
3
2 - 0
2
x
-2
-1
1
2
-
2 2
3
2 - 0
2
x
-2
-1
1
2
-
2 2
3
2 - 0
2
x
-2
-1
1
2
-
2 2
3
2 - 0
2
x
-2
-1
1
2
-
2 2
3
2 - 0
2
x
-2
-1
1
2
-
2 2
3
2 - 0
2
x
y
-2
-1
1
2
-
2 2
3
2 - 0
2
x
-2
-1
1
2
-2
23
2 - 0
2
x
-2
-1
1
2
-2
23
2 - 0
2
x
-2
-1
1
2
-2
23
2 - 0
2
x
-2
-1
1
2
-2
23
2 - 0
2
x
-2
-1
1
2
-2
23
2 - 0
2
x
y
-2
-1
1
2
-2
23
2 - 0
2
y y = sin = cos y y = sin = cos y y = sin = cos y y = sin = cos y y = sin = cos
sin 0
1.
yyyyyyyyyy
=........................ lim 0
0 0
sin lim lim sin
lim cos2.=........................ sin 0
1. =........................ lim 0
0 0
sin lim lim sin
lim cos2.=........................ sin 0
1. =........................ lim 0
0 0
sin lim lim sin
lim cos2.=........................ sin 0
1. =........................ lim 0
0 0
sin
1
0 1
A B
C
D
D
A 0
0
C
B
1
1
1
0 1
A B
C
D
D
A 0
0
C
B
1
1
1
0 1
A B
C
D
D
A 0
0
C
B
1
1
1
0 1
A B
C
D
D
A 0
0
C
B
1
1
1
0 1
A B
C
D
D
A 0
0
C
B
1
1
1
0 1
A B
C
D
D
A 0
0
C
B
1
1
1
0 1
A B
C
D
D
A 0
0
C
B
1
1
1
0 1
A B
C
D
D
A 0
0
C
B
1
1
1
0 1
A B
C
D
D
A 0
0
C
B
1
1
1
0 1
A B
C
D
D
A 0
0
C
B
1
1
0
B
D
0
B
D
0
B
D
0
B
D
0
B
D
0
B
D
0
B
D
0
B
D
0
B
D
0
B
D
Sandwich Theorem Squeeze Theorem
f, g h Ia I , f(a), g(a), h(a)
f(x) g(x) h(x) x I a x 6= a
limxa
f(x) = L = limxa
h(x) limxa
g(x) = L
Sandwich Theorem Squeeze TheoremSandwich Theorem Squeeze TheoremSandwich Theorem Squeeze TheoremSandwich Theorem Squeeze Theorem
f, g h I a I , f(a), g(a), h(a) a I , f(a), g(a), h(a)
f(x) g(x) h(x) x I a x 6= a
limxa
f(x) = L = limxa
h(x) limxa
g(x) = L
f(x) g(x) h(x) x I ax 6= a
limxa
f(x) = L = limxa
h(x) limxa
g(x) = L
-
krupornchaikrupornchaikrupornchaikrupornchai
1111....
0x lim
sin x = 0
2222.... 0x
lim
cos x = 1
3333.... 0x
lim
xxsin = 1 = = 1 = = 1 = = 1 =
0x lim
xsinx
4444.... 0x
lim
xxcos1 = 0 = = 0 = = 0 = = 0 =
0x lim
x1 - xcos
3 1. 2. lim 1 3. sine cos 4. 1 cosx , cosx 1 , 1-sec x secx 1 4
1111....
0x lim
sin x = 0
2222.... 0x
lim
cos x = 1
3333.... 0x
lim
xxsin = 1 = = 1 = = 1 = = 1 =
0x lim
xsinx
4444.... 0x
lim
xxcos1 = 0 = = 0 = = 0 = = 0 =
0x lim
x1 - xcos
3 1. 2. lim 1 3. sine cos 4. 1 cosx , cosx 1 , 1-sec x secx 1 4
1111....
0x lim
sin x = 0
2222.... 0x
lim
cos x = 1
3333.... 0x
lim
xxsin = 1 = = 1 = = 1 = = 1 =
0x lim
xsinx
4444.... 0x
lim
xxcos1 = 0 = = 0 = = 0 = = 0 =
0x lim
x1 - xcos
3 1. 2. lim 1 3. sine cos 4. 1 cosx , cosx 1 , 1-sec x secx 1 4
1111....
0x lim
sin x = 0
2222.... 0x
lim
cos x = 1
3333.... 0x
lim
xxsin = 1 = = 1 = = 1 = = 1 =
0x lim
xsinx
4444.... 0x
lim
xxcos1 = 0 = = 0 = = 0 = = 0 =
0x lim
x1 - xcos
3 1. 2. lim 1 3. sine cos 4. 1 cosx , cosx 1 , 1-sec x secx 1 4
1. 0x
lim
x8x5sin
2.
0x lim
xx3sinx5sin +
3. 0x
lim
2xx2cos1
4. 0x
lim
xsecxx2tan2x2
5. 0x
lim
x3tanx2
6. 0x
lim
x3sinx7tan
7. x
lim
x)xsin(
8. 0x
lim 1xsin3x2sin2
1xsinx2sin2+
+
9. f(x) = sinx 0x
lim
x)x(f)xx(f
+
10. f(x) = cosx 0x
lim
x)x(f)xx(f
+
11.
3x lim
)xcos(xtan3x3tan
+
12. 0x
lim
1xsec2x
13.x lim xcosx
xsinx++
14. x lim
x11x1cos
+
15. x lim x
12 sinx
1. 0x
lim
x8x5sin
2.
0x lim
xx3sinx5sin +
3. 0x
lim
2xx2cos1
4. 0x
lim
xsecxx2tan2x2
5. 0x
lim
x3tanx2
6. 0x
lim
x3sinx7tan
7. x
lim
x)xsin(
8. 0x
lim 1xsin3x2sin2
1xsinx2sin2+
+
9. f(x) = sinx 0x
lim
x)x(f)xx(f
+
10. f(x) = cosx 0x
lim
x)x(f)xx(f
+
11.
3x lim
)xcos(xtan3x3tan
+
12. 0x
lim
1xsec2x
13.x lim xcosx
xsinx++
14. x lim
x11x1cos
+
15. x lim x
12 sinx
1. 0x
lim
x8x5sin
2.
0x lim
xx3sinx5sin +
3. 0x
lim
2xx2cos1
4. 0x
lim
xsecxx2tan2x2
5. 0x
lim
x3tanx2
6. 0x
lim
x3sinx7tan
7. x
lim
x)xsin(
8. 0x
lim 1xsin3x2sin2
1xsinx2sin2+
+
9. f(x) = sinx 0x
lim
x)x(f)xx(f
+
10. f(x) = cosx 0x
lim
x)x(f)xx(f
+
11.
3x lim
)xcos(xtan3x3tan
+
12. 0x
lim
1xsec2x
13.x lim xcosx
xsinx++
14. x lim
x11x1cos
+
15. x lim x
12 sinx
1. 0x
lim
x8x5sin
2.
0x lim
xx3sinx5sin +
3. 0x
lim
2xx2cos1
4. 0x
lim
xsecxx2tan2x2
5. 0x
lim
x3tanx2
6. 0x
lim
x3sinx7tan
7. x
lim
x)xsin(
8. 0x
lim 1xsin3x2sin2
1xsinx2sin2+
+
9. f(x) = sinx 0x
lim
x)x(f)xx(f
+
10. f(x) = cosx 0x
lim
x)x(f)xx(f
+
11.
3x lim
)xcos(xtan3x3tan
+
12. 0x
lim
1xsec2x
13.x lim xcosx
xsinx++
14. x lim
x11x1cos
+
15. x lim x
12 sinx
1. 0x
lim
x8x5sin
2.
0x lim
xx3sinx5sin +
3. 0x
lim
2xx2cos1
4. 0x
lim
xsecxx2tan2x2
5. 0x
lim
x3tanx2
6. 0x
lim
x3sinx7tan
7. x
lim
x)xsin(
8. 0x
lim 1xsin3x2sin2
1xsinx2sin2+
+
9. f(x) = sinx 0x
lim
x)x(f)xx(f
+
10. f(x) = cosx 0x
lim
x)x(f)xx(f
+
11.
3x lim
)xcos(xtan3x3tan
+
12. 0x
lim
1xsec2x
13.x lim xcosx
xsinx++
14. x lim
x11x1cos
+
15. x lim x
12 sinx
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