limit theorem

1. Evaluating Limits of Algebraic Functions
by
Using the limit Theorems

2. Limit Theorem 1.
Limit of a Linear Function
If m and b are any
constants,

3. Example
lim
𝑥→2
(3𝑥 + 5)

4. Limit Theorem 2
Limit of a constant
If c is any constant then for
any number a

5. Example
lim
𝑥→2
−5

6. Limit Theorem 3
Limit of the Identity Function

7. Example
lim x
𝑥→2

8. Limit Theorem 4
Limit of the sum or difference of two Functions

9. Example
lim
𝑥→−2
(x2 -3x)=

10. Limit Theorem 5
Limit of the Sum or Diffrence of n Functions

11. Example
lim
𝑥→2
(x3+ 2x2 -6x -7)

12. Limit Theorem 6
Limit of the product of two Functions

13. Example
lim
𝑥→2
3x2 ∗ 2x

14. Limit Theorem 7
Limit of the product of n function

15. Example
lim
𝑥→2
(3x2 ∗ x ∗ 6)(-2x)

16. Limit Theorem 8
Limit of the nth power of a Function

17. Example
lim
𝑥→2
(3x2-x +6)2

18. Limit Theorem 9
limit of the Quotient of two function
If lim
𝑥→𝑎
𝑓 𝑥 = 𝐿 𝑎𝑛𝑑 lim
𝑥→𝑎
𝑔 𝑥 = 𝑚 𝑡ℎ𝑒𝑛 lim
𝑥→𝑎
𝑓(𝑥)
𝑔(𝑥)
=
𝐿
𝑀
, 𝑚 ≠ 0

19. Example
1. lim
𝑥→0
4𝑥 −5
2𝑥+1
2. lim
𝑥→3
𝑥2 −9
𝑥 −3

20. Limit Theorem 10
Limit of the nth root of a function
If n is a positive integer and
lim
𝑥→𝑎
𝑓 𝑥 = 𝐿 𝑡ℎ𝑒𝑛 lim
𝑥→𝑎
𝑛
𝑓(𝑥) =
𝑛
𝐿 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒
𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑡ℎ𝑎𝑡 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 𝐿 ≥ 0

21. Example
1.lim
𝑥→0
3
2𝑥2 + 5𝑥 − 3
2.lim
𝑥→1`
𝑥2−2𝑥+3
3𝑥 −1
3.lim
𝑥→4
𝑥−2
𝑥−4

22. Limit Theorem 11
Infinity Theorem
If r is any positive integers then
𝑖 lim
𝑥→0+
1
𝑥𝑟
= +∞
(ii) lim
𝑥→0−
1
𝑥𝑟 =
+∞ 𝑖𝑓 𝑟 𝑖𝑠 𝑒𝑣𝑒𝑛
−∞ 𝑖𝑓 𝑟 𝑖𝑠 𝑜𝑑𝑑

23. Example
1. lim
𝑥→0+
1
𝑋5
2. lim
𝑥→0−
1
𝑋100
3. lim
𝑥→0−
1
𝑋3

24. Limit Theorem 12
If a is any real number and as
lim
𝑥→𝑎
𝑓 𝑥 = 𝑐, 𝑤ℎ𝑒𝑟𝑒 𝑐 𝑖𝑠 𝑎 𝑛𝑜𝑛 𝑧𝑒𝑟𝑜 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑛𝑑 lim
𝑥→𝑎
𝑔 𝑥 = 0, 𝑡ℎ𝑒𝑛
(i) If c > 0 and g(x) > 0 through positive values of g(x)
𝐥𝐢𝐦
𝒙→𝒂
𝒇(𝒙)
𝒈(𝒙)
= +∞
(ii) If c > 0 and g(x) > 0 through negative values of g(x)
𝐥𝐢𝐦
𝒙→𝒂
𝒇(𝒙)
𝒈(𝒙)
= −∞
(iii) If c < 0 and g(x) > 0 through positive values of g(x)
𝐥𝐢𝐦
𝒙→𝒂
𝒇(𝒙)
𝒈(𝒙)
= −∞
(iv) If c < 0 and g(x) > 0 through negative values of g(x)
𝐥𝐢𝐦
𝒙→𝒂
𝒇(𝒙)
𝒈(𝒙)
= +∞

25. Seatwork 3._
1. lim
𝑥→−3
2𝑥+5
𝑥 −2
2. lim
𝑥→5
2𝑥2+5𝑥+2
𝑥2 −16
3. lim
𝑥→
1
2
𝑥3+8
𝑥2+𝑥+4
Directions: simplify the following limits and find their
boundaries.

26. Limit of Infinity
(i) lim
𝑥→−∞
1
𝑥𝑛 = 0
(ii) lim
𝑥→∞
1
𝑥𝑛 = 0

27. Example
1. lim
𝑥→∞
2𝑥+5
𝑥 −2
2. lim
𝑥→∞
2𝑥2+5𝑥+2
𝑥2 −16
3. lim
𝑥→∞
𝑥3+8
𝑥2+𝑥+4

28. Seatwork 3._
1. lim
𝑥→∞
𝑥+125
𝑥2 −144
2. lim
𝑥→∞
𝑥2 + 𝑥+27
𝑥2+2𝑥 + 1
3. lim
𝑥→∞
𝑥3+2𝑥2+𝑥 +27
𝑥2− 4
Directions: simplify the following limits and find their
boundaries.