Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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1:44 minutes

Problem 2.4.47

Textbook Question

Theory and Examples

Once you know limx→a+ f(x) and limx→a− f(x) at an interior point of the domain of f, do you then know limx→a f(x)? Give reasons for your answer.

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To determine \( \lim_{x \to a} f(x) \), we need to consider both the right-hand limit \( \lim_{x \to a^+} f(x) \) and the left-hand limit \( \lim_{x \to a^-} f(x) \).

The limit \( \lim_{x \to a} f(x) \) exists if and only if both the right-hand limit and the left-hand limit exist and are equal.

If \( \lim_{x \to a^+} f(x) = L \) and \( \lim_{x \to a^-} f(x) = L \), then \( \lim_{x \to a} f(x) = L \).

If \( \lim_{x \to a^+} f(x) \neq \lim_{x \to a^-} f(x) \), then \( \lim_{x \to a} f(x) \) does not exist.

Therefore, knowing both \( \lim_{x \to a^+} f(x) \) and \( \lim_{x \to a^-} f(x) \) allows us to determine \( \lim_{x \to a} f(x) \) only if they are equal.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits from the Right and Left

In calculus, the limit of a function as it approaches a point from the right (denoted as limx→a+ f(x)) and from the left (denoted as limx→a− f(x)) are crucial for understanding the behavior of the function near that point. If both one-sided limits exist and are equal, it suggests that the overall limit at that point may also exist.

Existence of a Limit

A limit exists at a point if the values of the function approach a specific number as the input approaches that point from both sides. For limx→a f(x) to exist, it is necessary that both limx→a+ f(x) and limx→a− f(x) not only exist but also be equal. If they differ, the overall limit does not exist.

Continuity and Discontinuity

A function is continuous at a point if the limit as it approaches that point equals the function's value at that point. Discontinuities can arise when the one-sided limits are not equal or when either limit does not exist. Understanding these concepts helps in determining the nature of the function at the point in question.

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Related Practice

Textbook Question

Use the formal definitions from Exercise 97 to prove the limit statements in Exercises 98–102.

lim x→2⁻ (1 / (x − 2)) = −∞

Textbook Question

Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x→⁻∞ (³√x − ⁵√x) / (³√x + ⁵√x)

Textbook Question

Limits as x → ∞ or x → −∞

lim x→∞ √(x² + 1) / (x + 1)

Textbook Question

Limits as x → ∞ or x → −∞

lim x→∞ (x − 3) / √(4x² + 25)

Textbook Question

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.

Textbook Question

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.

Textbook Question

Finding Limits

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

h. lim x → ∞ f(x)

Textbook Question

[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

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Theory and ExamplesOnce you know limx→a+ f(x) and limx→a− f(x) at an interior point of the domain of f, do you then know limx→a f(x)? Give reasons for your answer.

Key Concepts

Limits from the Right and Left

Existence of a Limit

Continuity and Discontinuity

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Theory and Examples

Once you know limx→a+ f(x) and limx→a− f(x) at an interior point of the domain of f, do you then know limx→a f(x)? Give reasons for your answer.