Lesson 2-3 The Laws of Limits

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

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

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. 7. 8. 9. 10. 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

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

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

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

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

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

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

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

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

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

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 111-113: [Day 1] 1, 3, 6, 10, 11, 13, 20 [Day 2] 33, 40, 41, 52