Calculus class one

1. 1
CHAPTER ONE
LIMITS AND FUNCTIONS
Case 1: Most limiting values can be evaluated by direct
substitution.
(1) Evaluate lim (x2 + 5x + 10)
x 5
Solution Lim x2
+ 5x + 10
x 5
= 52 + 5(5) + 10
= 60
(2) Evaluate lim Sin + Cos 2
6
Solution
But = 1800
So, = 180 = 300
6 6
Therefore, = 300
6
Meaning that 300
Note: In radian measurement, = 1800
So, the intended question is that:

2. 2
Evaluate lim Sin + Cos 2
300
= Sin 300 + Cos 2(300)
= Sin 300 + Cos 600
= +
= 1
(3) Evaluatelim log8x + logx8
logxx 4 32
= log84 + log48
log432
Note: From the laws of logarithm,
Logab = logb , logyz = logz
loga logy
So, applying the law above in our present solution
below, we will have this: = log84 + log48
log432
= log4 + log8 Note: log84 = log4
log8 log4 log8
log32

3. 3
log4 log48 = log 8
log4
= log223 + log223 log432 = log32 log2
log2 log4 log25
log22
= 2log2 + 3log2 3 2
= 5
2
Note:
log22 = 2log2, log25 =
5log2
log23 = 3log2, Just
like logab = bloga
=
+
= 4 +
9
6
= 13 5
6 2
= 13 x 2 = 13
6 5 15
= 2 + 3
2log2
2log2 + 3log2=
5log2
2log2
3log2 + 2log2
2
3
3
2
5
2
5
2

4. 4
(4) Evaluate
lim x3 + 2x2 + 2
Solution:
x 2
= 23 + 2(2)2 + 2
= 8 + 2(4) + 2
= 8 + 8 + 2
= 18
(5) Evaluate
lim 3x4 - 2x2 - 8
x -2
Solution:
lim 3x4 - 2x2 - 8
x -2
= 3(-2)4 - 2(-2)2 - 8
= 3(16)- 2(4)- 8
= 48 - 8 - 8 = 32
(6) Evaluate
lim log2 x+ logx8
x 8
= log28 + log88
= log 28 + log88

5. 5
= log8 log8
log2 + log8 =
log 23 log23
log2 + 3log23
= 3log2 +
3log2 log2 +
3log2 = 3 + 1
= 4
Explanation:
In the above, I applied the law of
logarithm, logab = logb loga
An example is:
log28 = log8
log2
Another example is:
Log416 = log16
log4
(7) Evaluate
lim log3x + logx27
x 81
Solution: lim log3x +
logx27 x 81
= lim log381 + log8127

6. 6
= log81 + log27
log3 log81
= log34 + log33
log31 log34
= 4log3 + 3log3
1log3 4log3
= 4 + 3
1 4
= 16 + 3 = 19
4 4

7. 7
+
2 2
2
2
(8) Evaluate
lim Cos + Sin
450
Solution:
lim Cos + Sin
450
= Cos 450 + Sin 450
= 2 + 2
= 2
= 2 2 Note: 0
2 Sin 45 =
= 2 2
2
= 2
(9) Evaluate
lim tan + Sin 2
450
Solution:
tan 450 + Sin 2 (450)
= 450
= tan 450 + Sin 2(450)
= tan 450 + Sin 900
= 1 + 1
= 2
Note: In the above solution, tan 450 = 1, Sin 900 =
1
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