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?

c. limx→0− f(x) = 0

Finding Limits Graphically

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

e. limx→0 f(x) exists

Finding Limits Graphically

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

d. limx→1− f(x) = 2

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.

lim ( 1 / x²/³ + 2 / (x − 1)²/³ ) as

a. x → 0⁺

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.

lim ( 1 / x¹/³ − 1 / (x − 1)⁴/³ ) as

a. x → 0⁺

Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

y = 1/(x − 1)

Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

y = −3/(x − 3)

Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = x³ / (x³ − 8)

Use the graph of the greatest integer function y = ⌊x⌋, Figure 1.10 in Section 1.1, to help you find the limits in Exercises 21 and 22.

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b. limt→4−(t−⌊t⌋)

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)

lim x → ±∞ f(x) = 0, lim x → 2⁻ f(x) = ∞, and lim x → 2⁺ f(x) = ∞

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)

lim x → ±∞ k(x) = 1, lim x → 1⁻ k(x) = ∞, and lim x → 1⁺ k(x) = −∞

Finding Limits of Differences When x → ±∞

Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)

lim x → ∞ (√(x + 9) − √(x + 4))

Finding Limits of Differences When x → ±∞

Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)

lim x → −∞ (√(x² + 3) + x)

Finding Limits of Differences When x → ±∞

Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)

lim x → ∞ (√(9x² − x) − 3x)

Using the Formal Definitions

Use the formal definitions of limits as x → ±∞ to establish the limits in Exercises 91 and 92.

If f has the constant value f(x) = k, then lim x → ∞ f(x) = k.