1. Limits and Continuity

Finding Limits Graphically

Which of the following statements about the function y = f(x) graphed here are true, and which are false?

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k. limx→3+ f(x) does not exist.

Finding Limits Graphically

Let f(x) = {3 - x , x < 2

2, x = 2

x/2, x > 2

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a. Find limx→2+ f(x), limx→2− f(x), and f(2).

Finding Limits Graphically

Let f(x) = {3 - x , x < 2

2, x = 2

x/2, x > 2

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c. Find limx→−1− f(x) and limx→−1+ f(x).

Finding Limits Graphically

Graph the functions in Exercises 9 and 10. Then answer these questions.

f(x) = {x,−1 ≤ x < 0, or 0 < x ≤ 1

1, x = 0

0, x < −1 or x > 1

d. At what points does the right-hand limit exist but not the left-hand limit?

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