limit pada ketakhinggaan

Limit pada
Ketakhinggaan
lim
Untuk semua n > 0,
Dan diberikan
Ex. xlim
→∞
=
x →∞
x→∞
x →∞
( x ) − lim 4
⎛
⎞
4 x 2 5 x 21
− 3+ 3
⎜
⎟
3
x
= lim ⎜ 3 x 2 x
⎟
x →∞ 7 x
5 x 10 x 1 ⎟
⎜
⎜ 3 + 3 − 3 + 3⎟
x
x
x ⎠
⎝ x
5
21
⎛ 4
− 2 + 3
⎜
x
= lim ⎜ x x
x→ ∞
5 10
1
⎜7+ − 2 + 3
x x
x
⎝
=0
⎞
⎟
⎟
⎟
⎠
⎛ 2 x3 3x 2 2
− 3 + 3
⎜
3
x
= lim ⎜ 3 x 2 x
x →∞ x
⎜ − x − 100 x + 1
⎜ 3
x3
x3
x3
⎝x
2
2
0−4
=
4
x →∞
3.
⎞
⎟
⎟
⎟
⎟
⎠
3 2
⎛
⎞
⎜ 2 − x + x3
⎟
= lim ⎜
x→∞
1 100 1 ⎟
⎜1− − 2 + 3 ⎟
x x
x ⎠
⎝
Bagi
dgn x 2
( x ) + lim ( 1 x ) = 3 + 0 + 0 = − 3
lim 3 + lim 5
x →∞
⎛ 4 x 2 − 5 x + 21 ⎞
lim ⎜ 3
⎟
x →∞ 7 x + 5 x 2 − 10 x + 1
⎝
⎠
0
7
1
1
= lim
=0
x n x →−∞ x n
3+ 5 + 1 2
3x 2 + 5 x + 1
x
x
= lim
2
2 −4
x
→∞
2 − 4x
2
x
lim 2
=
⎛ 2 x3 − 3x2 + 2 ⎞
1. lim ⎜ 3
⎟
2
x→ ∞
⎝ x − x − 100 x + 1 ⎠
1
terdefinisi.
xn
x →∞
2.
Contoh lain
⎛ x2 + 2x − 4 ⎞
lim ⎜
⎟
x →∞
⎝ 12 x + 31 ⎠
⎛ x2 2 x 4 ⎞
− ⎟
⎜ +
x x⎟
= lim ⎜ x
x →∞
⎜ 12 x + 31 ⎟
⎜ x
x ⎟⎠
⎝
4⎞
⎛
⎜ x+2− x ⎟
= lim ⎜
⎟
x →∞
⎜ 12 + 31 ⎟
x ⎠
⎝
∞ + 2
=
12
=∞
4.
⎛
= lim ⎜⎜
x →∞
⎜
⎝
lim
x →∞
(
(
x2 + 1 − x
x2 + 1 − x
1
)
2
=2
1
)
⎞
x2 + 1 + x ⎟
⎟
x2 + 1 + x ⎟
⎠
⎛ x2 + 1 − x2 ⎞
= lim ⎜
⎟
2
x →∞
⎝ x +1 + x ⎠
⎛
⎞
1
= lim ⎜
⎟
2
x →∞
x
+
+
x
1
⎝
⎠
1
1
=
= =0
∞+∞ ∞
1
Limit Tak Hingga
Contoh
20
Untuk semua n > 0,
15
Temukan limitnya
10
lim+
x→a
1
( x − a)
n
5
=∞
-8
-6
-4
-2
2
-5
-10
1.
-15
1
lim−
=∞
x →a ( x − a ) n
⎛ 3x2 + 2 x + 1 ⎞
lim+ ⎜
⎟
x →0
2x2
⎝
⎠
-20
⎛ 2x +1 ⎞
⎛ 2x +1 ⎞
lim ⎜
⎟ = lim
2 x + 6 ⎠ x→−3+ ⎜⎝ 2( x + 3) ⎟⎠
40
2.
30
20
x →−3+ ⎝
10
jika n genap
-2
2
4
⎛ 3+ 2 + 1 2 ⎞
x
x ⎟
= lim+ ⎜
⎟⎟
x →0 ⎜
2
⎜
⎝
⎠
=
3+∞ + ∞
=∞
2
= −∞
40
6
-10
lim −
x→ a
1
= −∞
( x − a)n
20
-20
20
15
10
-8
-6
-4
-2
2
5
-20
-8
-6
-4
-2
2
-5
jika n ganjil
-10
-15
20
Tangent and Secant
Limit dan Fungsi Trigonometri
Tangent dan secant kontinu disemua titik kecuali
Gambar dari fungsi trig memberikan
x ≠±π , ±3π , ±5π , ±7π ,L
2
2
2
2
f ( x) = sin x and g ( x) = cos x
y = sec x
1
1
15
y = tan x
0.5
0.5
30
10
20
5
10
-10
-5
5
10
-10
-5
5
10
-6
-6
-0.5
-0.5
-4
-2
2
4
6
-4
-2
2
4
6
-5
-10
-10
-20
-1
-1
-15
-30
Jadi fungsinya kontinu pada sebarang titik
lim sin x = sin c and lim cos x = cos c
x →c
x →c
2
Limit dan Fungsi Exponential
Contoh
a) lim + sec x =
( 2)
x→ π
c)
lim
(
x → −3π
2
)
+
−∞
tan x = −∞
e) lim− cot x = −∞
b)
lim − sec x
( 2)
x→ π
d)
(
lim cot x =
(
x → −3π
2
)
lim
(
x → −3π
10
10
6
2
x →π
cos x
2 ) sin x
=
)
−
tan x =
∞
4
0
=0
1
4
4
2
2
-6
-4
-2
2
4
• Garis y = L disebut asimtot horisontal pada
kurva y = f(x) jika salah satu dibawah ini benar
lim f ( x) = L or lim f ( x) = L.
x →−∞
-6
6
x →c
x →c
2
4
6
Gambar diatas menunjukkan bahwa fungsi
exponential kontinu disemua.
lim a x = a c
x →c
Contoh
Tentukan Asimtot fungsi berikut
x2 + 1
x2 −1
(i) lim− f ( x) = −∞
1. f ( x) =
(iii) lim f ( x) = 1.
x →∞
Shg garis y = 1 adl asimtot
horisontal
Shg garis x = 1 adl asimtot
vertical
(ii) lim− f ( x) = +∞.
x →−1
lim− f ( x) = ±∞ or lim+ f ( x) = ±∞.
-2
-2
x →1
• Garis x = c disebut asimtot vertikal pada kurva
y = f(x) jika salah satu dibawah ini benar
-4
-2
Asimptot
x →∞
y = ax , 0 < a < 1
8
8
f) lim tan x = 1
x →π
g)
y = a , a >1
6
lim
x → −3π
=∞
x
10
7.5
5
2.5
-4
-2
2
4
-2.5
Shg garis x = -1 adl asimtot
vertical
-5
-7.5
-10
3
2.
f ( x) =
x −1
x2 − 1
⎛ x −1 ⎞
(i) lim f ( x) = lim ⎜ 2 ⎟
x →1
x →1
⎝ x −1 ⎠
⎛
⎞
x −1
⎛ 1 ⎞ 1
= lim ⎜
⎟ = lim ⎜
⎟= .
x →1
⎝ ( x − 1)( x + 1) ⎠ x →1 ⎝ x + 1 ⎠ 2
(iii) lim f ( x) = 0.
x →∞
Shg garis y = 0 adl
asimtot horisontal
Shg garis x = 1 bukan
merupakan asimtot vertical
(ii) lim+ f ( x) = +∞.
x →−1
10
7.5
5
2.5
-4
-2
2
4
-2.5
-5
Shg garis x = -1 adl asimtot
vertical
-7.5
-10
4