Definition of
Trigonometric functions
Six trigonometric value
180 = π
Radian)
𝜋
90 =
2
𝑏
𝑐
𝑎
cos 𝜃 =
𝑐
sin 𝜃 =
b: opposite
θ
a: adjacent
𝑏
𝑎
𝑎
cot 𝜃 =
𝑏
tan 𝜃 =
360 = 2𝜋 in Radian
(sin 𝜃) + (cos 𝜃) = 1
(tan 𝜃) + 1 = (sec 𝜃)
(cot 𝜃) + 1 = (csc 𝜃)
𝜃(in Radian)
0 =0
sec 𝜃 =
𝑐
𝑎
csc 𝜃 =
𝑐
𝑏
𝜃 = 30 =
1
45 =
θ
𝑏
𝑎
𝑎
cot 𝜃 =
𝑏
tan 𝜃 =
csc 𝜃 =
𝑐
𝑏
𝜃 = 60 =
θ
𝑏
𝑐
𝑎
cos 𝜃 =
𝑐
𝑐
𝑎
60 =
360
2𝜋 𝜋
=
=
6
6
3
45 =
360
2𝜋 𝜋
=
=
8
8
4
30 =
360
2𝜋 𝜋
=
=
12
12 6
0 =0
sin 𝜃 =
sec 𝜃 =
(in
5
θ
90 =
180 = π
Section 2-1 Limit and
Example
Exercise
Limit laws
lim𝑥 + 1 = 2
Definition: The limit of f
→
at a is L, is defined by
lim2𝑥 + 1 =
→
𝑥 −1
=
→ 𝑥−1
lim
lim 𝑓(𝑥) = 𝐿
→
lim
→
= lim 𝑓(𝑥) = lim 𝑓(𝑥)
𝑥 − 25
=
𝑥−5
→
→
(Two sided limit)
One-sided limit:
lim 𝑓(𝑥): Right hand
→
limit
lim 𝑓(𝑥): left hand
→
limit
where 𝑘 is a constant
Section 2-2 Limit
lim 𝑘 = 𝑘
Theorems
lim 𝑥 = 𝑎
1. Constant law
2. Identity law
3. Constant multiple
lim 𝑘𝑓(𝑥) = 𝑘lim 𝑓(𝑥), constant multiple
4. Sum of limits
lim (𝑓(𝑥) + 𝑔(𝑥))= lim 𝑓(𝑥) + lim 𝑔(𝑥)
5. Difference of limits
lim (𝑓(𝑥) − 𝑔(𝑥))= lim 𝑓(𝑥) − lim 𝑔(𝑥)
6. Multiplication of
lim(𝑓(𝑥)𝑔(𝑥))= lim 𝑓(𝑥)lim 𝑔(𝑥)
→
→
limits
7. Division of limits
→
→
→
→
→
→
→
lim
→
→
→
( )
( )
= →
→
→
→
( )
( )
lim (𝑓(𝑥)) = (lim 𝑓(𝑥))
8. n-power
9. nth-root
→
→
lim 𝑓(𝑥) =
lim 𝑓(𝑥) ,lim 𝑓(𝑥) ≥ 0 , if n is even
→
1. Constant law
lim 𝑘 = 𝑘
→
→
lim 5 = 5
lim 4 =
lim 𝑥 = 2
lim 𝑥 =
lim 7𝑥 = 7(2) = 14
lim 6𝑥 =
→
→
→
2. Identity law
lim 𝑥 = 𝑎
→
→
→
3. Constant multiple
lim 𝑓(𝑥) = 𝐿
→
→
→
Then lim 𝑘𝑓(𝑥) = 𝑘𝐿
→
Suppose lim 𝑓(𝑥) = 𝐿 and lim 𝑔(𝑥) = 𝑀 (𝐿 and 𝑀 exist as finite numbers. )
→
4. Sum
lim 𝑓(𝑥) = 𝐿 and
→
lim 𝑥 + 5 = 2 + 5 = 7
lim 𝑥 + 4 =
→
→
→
6
lim 𝑔(𝑥) = 𝑀
→
Then lim 𝑓(𝑥) + 𝑔(𝑥) =
→
𝐿+𝑀
5. Subtract
lim 𝑓(𝑥) − 𝑔(𝑥) = 𝐿 − 𝑀
lim(𝑥 − 5) = 2 − 5 = −3
lim 𝑥 − 4 =
lim 𝑥 = 2 = 4
lim6𝑥 =
lim
𝑥 − 5 −3
=
→
𝑥
4
lim
lim(𝑥 − 5)3 = (2 − 5) = −27
lim(𝑥 − 4)3 =
lim
lim 6𝑥 + 10 =
→
→
→
6. Multiplication
lim 𝑓(𝑥)𝑔(𝑥) = 𝐿𝑀
→
→
→
7. Division
lim
→
( )
( )
=
,if 𝑀 ≠ 0
8. Power
lim (𝑓(𝑥)) = 𝐿
𝑥−4
=
→ 6𝑥
→
→
→
9. Radical
−16𝑥 = lim
→
(−16 ∗ 4) = −4
→
→
lim 𝑓(𝑥) = lim (𝑓(𝑥))
→
→
= √𝐿
lim 2𝑥 + 1 = lim √9 = 3
→
lim 3𝑥 + 4 =
→
→
L>0 if n is even
10. indeterminate form ( )
|𝑥|
=
→ 𝑥
lim
|𝑥 − 1|
𝑥
=
lim = 1, 𝑥 >lim
→ 𝑥−1
→ 𝑥
−𝑥
lim
= −1, 𝑥
→
𝑥
𝑥−1
lim
= 1, 𝑥 > 1
→ 𝑥−1
−(𝑥 − 1)
lim
= −1, 𝑥 < 1
→
𝑥−1
|1 − 𝑥|
=
→ 𝑥−1
lim
lim
|1 − 𝑥|
=
𝑥−1
lim
|1 − 𝑥|
=
𝑥−1
→
→
11. Factorization:
indeterminate form ( )
lim
→
(𝑥 + 2)(𝑥 − 2𝑥 + 4)
𝑥 +8
= lim
→
𝑥+2
𝑥+2
→
→
= lim (𝑥 − 2𝑥 + 4) = 4 − 2(−2) + 4
𝑥 −4
→ 𝑥−2
→
lim
= lim
lim
lim
= 12
→
=
𝑥 −9
𝑥 + 2𝑥 − 15
(𝑥 − 2)(𝑥 + 2)
𝑥−2
= lim𝑥 + 2 = 4
→
12. Rationalize: indeterminate
form ( )
√𝑥 + 9 − 3
→
𝑥
lim
lim
→
(√𝑥 + 9 − 3)(√𝑥 + 9 + 3)
√𝑥 + 9 − 3
= lim
→
𝑥
𝑥 (√𝑥 + 9 + 3)
= lim
→
= lim
→
(𝑥 + 9) − 9
𝑥 (√𝑥 + 9 + 3)
1
√𝑥 + 9 + 3
7
=
1
6
√𝑥 + 4 − 2
→
𝑥
lim
Rationalization
lim
→
Section 2.4
√9 + 𝑥 − 3
𝑥
lim
→
sin 𝑥
=1
→
𝑥
sin 2𝑥
=
→
𝑥
lim
Trigonometric Limits
√5 + 𝑥 − √5
𝑥
lim
sin 2𝑥
=
→ sin 3𝑥
lim
lim
→
cos 𝑥 − 1
=0
𝑥
lim
=
→
indeterminate form ( )
lim
sin2 𝜃
=
3𝜃
lim
sin5 𝜃
=
sin3 𝜃
lim
tan5 𝜃
=
sin3 𝜃
lim
tan5 𝜃
=
𝜃
→
sin 𝜃
=1
→
𝜃
lim
-0.1
θ
sin 𝜃 0.9983
𝜃
-0.01
-0.001
0
0.001
0.01
0.1
0.99998
0.99999
X
0.99999
0.99998
0.99833
→
→
→
indeterminate form ( )
lim
→
(cos 𝜃 − 1)(cos 𝜃 + 1)
cos 𝜃 − 1
= lim
→
→
𝜃
𝜃(cos 𝜃 + 1)
lim
cos 𝜃 − 1
=0
𝜃
−(sin 𝜃)
→ 𝜃(cos 𝜃 + 1)
= lim
(− sin 𝜃) sin 𝜃
=0
→ 𝜃(cos 𝜃 + 1)
= lim
lim 𝑓(𝑥) = 𝐿
One sided limit
→
lim 𝑓(𝑥) = 𝐿
→
Section 2.5
lim 𝑓(𝑥) = ±∞
Limit involving
lim 𝑓(𝑥) = ±∞
Infinity
lim 𝑓(𝑥) = ±∞
→
→
lim 𝑓(𝑥) = ±∞
→
x
1
𝑥
-0.0001
0
0.0001
0.01
0.1
-10
-100
-10000
X
10000
100
10
=
lim
1
=
𝑥−1
lim
1
=
𝑥−1
lim
1
=
𝑥−1
lim
1
=
𝑥−1
→
→
lim
= 0,
→
does not exit
-0.01
=∞
→
lim
→
lim
→
= −∞
=0
:
:
1
𝑥−𝑎
lim
→
-0.1
lim
lim
→
Limit involving infinity
→
𝑥−2
𝑥 −4
→
→
lim
lim
x=0: is a vertical asymptote
→
y=0: is a horizontal asymptote
→
8