1. Limits and Continuity

Theory and Examples

a. If limx→0 f(x) / x² = 1, find limx→0 f(x).

Suppose limx→4 f(x) = 0 and lim x→4 g(x) = −3. Find

c. limx→4 (g(x))²

Using Limit Rules

Suppose lim x→0 f(x) = 1 and lim x→0 g(x) = −5. Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation.

limx→0 (2f(x) − g(x)) / (f(x) + 7)² = limx→0 (2f(x) − g(x)) / limx→0 (f(x) + 7)² (a)

(We assume the denominator is nonzero.)

(lim x→0 2f(x) − lim x→0 g(x)) / (lim x→0 (f(x) + 7))² (b)

= (2 lim x→0 f(x) − lim x→0 g(x)) / (lim x→0 f(x) + lim x→0 7)² (c)

= ((2)(1) − (−5)) / (1 + 7)² = 7/64

Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–20.

limx→1− (1/(x + 1))((x + 6)/x)((3 − x)/7)

Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–20.

limh→0− (√6 − √(5h² + 11h + 6))/ h

Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–20.

a. limx→1+ (√2x (x − 1)) / |x − 1|

Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–20.

a. limx→0+ (1 − cos x) / |cos x − 1|

Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.

lim x→(−π/2)⁺ sec x

Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.

lim θ→0 (2 − cot θ)

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim x/(x² − 1) as

d. x→−1⁻

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x²/2 − 1/x) as

b. x→0⁻

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x²/2 − 1/x) as

d. x→−1

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 1) / (2x + 4) as

b. x→−2⁻

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 3x + 2) / (x³ − 2x²) as

a. x→0⁺

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 3x + 2) / (x³ − 2x²) as

d. x→2