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

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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1:19 minutes

Problem 2.2.48

Textbook Question

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→0 (1 + x + sin x) / (3 cosx)

Verified step by step guidance

Step 1: Understand the problem. We need to find the limit of the function (1 + x + sin(x)) / (3 cos(x)) as x approaches 0.

Step 2: Apply the limit property that allows us to evaluate the limit of a quotient by finding the limits of the numerator and the denominator separately, provided the limit of the denominator is not zero.

Step 3: Evaluate the limit of the numerator, 1 + x + sin(x), as x approaches 0. Use the fact that sin(x) approaches 0 as x approaches 0, and x approaches 0 as x approaches 0. Therefore, the limit of the numerator is 1 + 0 + 0 = 1.

Step 4: Evaluate the limit of the denominator, 3 cos(x), as x approaches 0. Use the fact that cos(x) approaches 1 as x approaches 0. Therefore, the limit of the denominator is 3 * 1 = 3.

Step 5: Combine the results from Steps 3 and 4 to find the limit of the entire expression. The limit of the quotient is the limit of the numerator divided by the limit of the denominator, which is 1 / 3.

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Key Concepts

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

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches 0 helps determine the behavior of the function near that point.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. They play a crucial role in calculus, especially when dealing with limits, derivatives, and integrals involving angles. Understanding their behavior near specific points, like x = 0, is vital for solving limit problems.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) yields an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful when dealing with limits involving trigonometric functions.