1. Lesson 2-3
The Laws of Limits

2. Objectives Find Limits using the Laws of Limits
Understand and use the Squeeze Theorem

3. Vocabulary
Greatest Integer Function – [[ x ]], the largest integer that is less than or equal to x
Continuous – (studied in detail in 2.5) no interrupt or abrupt change in the function

4. Laws of Limits Summary lim f(x) lim g(x) lim [f(x) + g(x)] =
Suppose c is a constant and the and exist. Then --- 1. 2. 3. 4. 5. 6.
11.
lim f(x)
xa
lim g(x)
xa
lim [f(x) + g(x)] =
xa
lim f(x) +
xa
lim g(x)
xa
Sum Law
Difference Law
Constant Multiple Law
Product Law
Quotient Law
from Product
Special Limits
from 6 and 8
Generalization
lim [f(x) - g(x)] =
xa
lim f(x) -
xa
lim g(x)
xa
lim [cf(x)] =
xa
c lim f(x)
xa
lim [f(x) • g(x)] =
xa
lim f(x) •
xa
lim g(x)
xa
lim [f(x) / g(x)] =
xa
lim f(x) /
xa
lim g(x) , if
xa
lim g(x) ≠ 0
xa
lim [f(x)]ⁿ =
xa
[lim f(x)]ⁿ
xa
(n is a positive integer)
lim c = c
xa
lim x = a
xa
lim √x = √a
xa ⁿ
lim xⁿ = aⁿ
xa
lim √f(x) =
xa
√lim f(x) ⁿ

5. Squeeze Theorem
Theorem: If f(x) ≤ g(x) when x is near a (except possibly at a) and the limits of f and g exist as x approaches a, then
SQUEEZE THEOREM:
(Conditions:) 1) If f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a) and 2)
then
lim f(x) ≤
xa
lim g(x)
xa
lim f(x) =
xa
lim h(x) = L
xa y
y = x²
lim g(x) = L
xa
y = x² sin (1/x) x
Figure 8 from pg 111
y = -x²

6. Example 1 Find 87 Lim 4x² - 10x + 3 = Polynomials – just use
Direct Substitution – f(a)
Lim 4x² - 10x + 3 =
x→ 6

7. Example 2 Find x² - 6x + 5 Lim ------------------- = -4 x – 1
If direct substitution fails
(yields 0/0) try factoring
To algebraically simplify
x→ 1

8. Example 3 Find 4x + 44 Lim ------------ = 6x – 29 4
If direct substitution works,
then be happy!
x→ 5

9. Example 4 Find x² x ≥ 3 Lim f(x) = where f(x) = 6x – 4 x < 3 DNE
Since we are at a “break point”,
if direct substitution for the one-sided limits
yields different values, then the two-sided limit is DNE
x→ 3

10. Example 5 Find: Lim f(x) + Lim g(x) = Lim (f + g) (x) = 1 + 3 = 4
x→ x→ 2
= 4
Use laws of limits to simplify the problem
Lim (f + g) (x) =
x→ 2

11. Example 6 Find (1/x) - x Lim --------------- = 2 (1/x) - 1
If direct substitution yields 0/0,
then algebraically simplify the
expression.
x→ 1

12. Example 7 Find 2 + x - x Lim ------------------ = DNE x
If direct substitution yields 0/0,
then algebraically simplify the
expression. They don’t always exist!
x→ 0

13. Proof Find sin θ Lim ------------ = 1 θ
θ → 1
Since area of the inscribed ∆ ≤ area of the sector ≤ outer ∆
½ sin θ ≤ ½ θ ≤ ½ tan θ so (sin θ) / θ ≤ 1
Next multiply the second part by (2 cos θ) / θ and we get
cos θ ≤ (sin θ) / θ
Combine and take the limits
Lim cos θ ≤ lim (sin θ) / θ ≤ lim 1
1 ≤ lim (sin θ) / θ ≤ 1

14. Summary & Homework Summary:
Try to find the limit via direct substitution
Use algebra to simplify into useable form
Use laws of limits to simplify and solve
Use squeeze theorem
Homework: pg : [Day 1] 1, 3, 6, 10, 11, 13, 20 [Day 2] 33, 40, 41, 52