1. Limits and Continuity

Theory and Examples

Suppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.

Formal Definitions of One-Sided Limits

Greatest integer function Find (a) limx→400+ ⌊x⌋ and (b) limx→400− ⌊x⌋; then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about limx→400 ⌊x⌋? Give reasons for your answer.

Finding Limits

For the function f whose graph is given, determine the following limits. Write ∞ or −∞ where appropriate.

h. lim x → ∞ f(x)

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.

a. How does the graph behave as x → 0⁺?

Give reasons for your answers.

y = (3/2)(x − (1 / x))²/³

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.

c. How does the graph behave near x = 1 and x = −1?

Give reasons for your answers.

y = (3/2)(x − (1 / x))²/³

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (tan 2x) / x

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 (1 − cos θ) / sin 2θ

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

llimx→0 (x −x cos x) / sin² 3x

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0 sin(sin h) / sin h

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 sin θ cot 2θ

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 tan θ / θ²cot 3θ

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (cos²x − cos x) / x²

Graphing Simple Rational Functions

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

y = (x + 3)/(x + 2)

Graphing Simple Rational Functions

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

y = 2x/(x + 1)

Domains and Asymptotes

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

y = 2x / (x² − 1)