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
- 8. Definite Integrals3h 25m
1. Limits and Continuity
Finding Limits Algebraically
Problem 3.5.15
Textbook Question
Use Theorem 3.10 to evaluate the following limits.
lim xπ 0 (tan 5x) / x
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1
Identify the limit to evaluate: lim (x -> 0) (tan(5x) / x).
Recognize that as x approaches 0, both the numerator tan(5x) and the denominator x approach 0, creating an indeterminate form of 0/0.
Apply Theorem 3.10, which states that if lim (x -> a) f(x) = 0 and lim (x -> a) g(x) = 0, then lim (x -> a) (f(x)/g(x)) = lim (x -> a) (f'(x)/g'(x)) if the limit on the right exists.
Differentiate the numerator and denominator: f(x) = tan(5x) and g(x) = x. Find f'(x) = 5sec^2(5x) and g'(x) = 1.
Substitute x = 0 into the derivatives: lim (x -> 0) (5sec^2(5x) / 1) to evaluate the limit.
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