1. Limits and Continuity

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0⁺ 1 / 3x

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→−5⁻ (3x) / (2x + 10)

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 (−1) / (x² (x + 1))

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

b. lim x→0⁻ 2 / (3x¹/³)

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 4 / x²/⁵

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let G(x)=(x + 6)/(x² + 4x − 12)

b. Support your conclusions in part (a) by graphing G and using Zoom and Trace to estimate y-values on the graph as x→−6.

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let h(x)=(x² − 2x − 3)/(x² − 4x + 3)

b. Support your conclusions in part (a) by graphing h near c = 3 and using Zoom and Trace to estimate y-values on the graph as x→3.

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let F(x)=(x² + 3x + 2)/(2−|x|)

b. Support your conclusion in part (a) by graphing F near c = -2 and using Zoom and Trace to estimate y-values on the graph as x→−2.

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let g(θ) = (sinθ) / θ.

b. Support your conclusion in part (a) by graphing g near θ₀ = 0.

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let f(x)=(x² − 1)/(|x| − 1).

b. Support your conclusion in part (a) by graphing f near c = -1 and using Zoom and Trace to estimate y-values on the graph as x→−1.

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→3 (3x − 7) = 2

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→9 √(x − 5) = 2

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 √(4 − x) = 2

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→1 f(x) = 1 if f(x) = {x², x ≠ 1

2, x = 1

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→1 1/x = 1