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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Multiple Choice

Use series to evaluate the limit: $lim_{x \to 0} \frac{1 - \cos (2 x)}{1 + 2 x - e^{2 x}}$ .

0

1

2

\frac{1}{2}

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Verified step by step guidance

Step 1: Recognize that the problem involves evaluating a limit as x approaches 0. To simplify the expressions, expand both the numerator and denominator using Taylor series approximations around x = 0.

Step 2: Expand $ \cos(2x) $ using its Taylor series: $ \cos(2x) = 1 - \frac{(2x)^2}{2!} + \frac{(2x)^4}{4!} - \dots $. Subtract $ \cos(2x) $ from 1 to get $ 1 - \cos(2x) = \frac{(2x)^2}{2!} - \frac{(2x)^4}{4!} + \dots $.

Step 3: Expand $ e^{2x} $ using its Taylor series: $ e^{2x} = 1 + 2x + \frac{(2x)^2}{2!} + \frac{(2x)^3}{3!} + \dots $. Subtract $ e^{2x} $ from $ 1 + 2x $ to get $ 1 + 2x - e^{2x} = -\frac{(2x)^2}{2!} - \frac{(2x)^3}{3!} - \dots $.

Step 4: Substitute the Taylor series expansions for $ 1 - \cos(2x) $ and $ 1 + 2x - e^{2x} $ into the limit expression: $ \lim_{x \to 0} \frac{1 - \cos(2x)}{1 + 2x - e^{2x}} = \lim_{x \to 0} \frac{\frac{(2x)^2}{2!} - \frac{(2x)^4}{4!} + \dots}{-\frac{(2x)^2}{2!} - \frac{(2x)^3}{3!} - \dots} $.

Step 5: Simplify the fraction by canceling common terms in the numerator and denominator. Focus on the leading-order terms (the lowest power of x) in both the numerator and denominator, as higher-order terms become negligible as x approaches 0. This will allow you to evaluate the limit and determine the result.