Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 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
1. Limits and Continuity
Introduction to Limits
5:01 minutes
Problem 2.7.31
Textbook Question
Textbook QuestionUse the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.
lim x→−3 |2x|=6 (Hint: Use the inequality ∥a|−|b∥≤|a−b|, which holds for all constants a and b (see Exercise 74).)
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit Definition
The precise definition of a limit states that for a function f(x) to approach a limit L as x approaches a value c, for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This formalizes the intuitive idea of limits and is essential for proving limit statements rigorously.
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One-Sided Limits
Absolute Value and Inequalities
The absolute value function measures the distance of a number from zero on the number line. The inequality ∥a|−|b∥≤|a−b| helps in establishing relationships between the absolute values of expressions, which is crucial when manipulating limits involving absolute values, such as |2x| in this case.
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Initial Value Problems
Epsilon-Delta Relationship
In limit proofs, the relationship between ε and δ is critical. It specifies how close x must be to c (within δ) to ensure that f(x) is within ε of L. Establishing this relationship is key to demonstrating that the limit exists and is valid under the conditions set by the definition of a limit.
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Finding Differentials
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