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

Previous problemNext problem

1:51 minutes

Problem 2.2.49

Textbook Question

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→−π √(x + 4) cos(x + π)

Verified step by step guidance

First, understand the problem: We need to find the limit of the function \( \sqrt{x + 4} \cos(x + \pi) \) as \( x \) approaches \( -\pi \).

Substitute \( x = -\pi \) into the expression \( \sqrt{x + 4} \). This gives \( \sqrt{-\pi + 4} \). Calculate \( -\pi + 4 \) to find the value inside the square root.

Next, substitute \( x = -\pi \) into the expression \( \cos(x + \pi) \). This simplifies to \( \cos(-\pi + \pi) = \cos(0) \). Recall that \( \cos(0) = 1 \).

Combine the results from the previous steps: The limit expression becomes \( \sqrt{-\pi + 4} \times 1 \).

Evaluate \( \sqrt{-\pi + 4} \) to find the final limit value. Ensure that the expression inside the square root is non-negative, as the square root function is defined for non-negative values.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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 -π involves determining the behavior of the function near that point.

Trigonometric Functions

Trigonometric functions, such as 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 how these functions behave near specific points is key to solving limit problems.

Square Root Function

The square root function, denoted as √(x), is defined for non-negative values of x and represents the principal square root. In limit problems, it is important to consider the domain of the function and how it behaves as the input approaches certain values. In this case, √(x + 4) must be evaluated as x approaches -π to ensure the expression remains valid.