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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1:50 minutes

Problem 2.2.56a

Textbook Question

Suppose that limx→−2 p(x) = 4, limx→−2 r(x) = 0, and limx→−2 s(x) = −3. Find

a. limx→−2 (p(x) + r(x) + s(x))

Verified step by step guidance

Step 1: Understand the problem statement. We are given the limits of three functions as x approaches -2: lim(x→−2) p(x) = 4, lim(x→−2) r(x) = 0, and lim(x→−2) s(x) = −3. We need to find the limit of the sum of these functions as x approaches -2.

Step 2: Recall the property of limits that states the limit of a sum is the sum of the limits. This means that lim(x→−2) (p(x) + r(x) + s(x)) = lim(x→−2) p(x) + lim(x→−2) r(x) + lim(x→−2) s(x).

Step 3: Substitute the given limits into the equation from Step 2. We have lim(x→−2) p(x) = 4, lim(x→−2) r(x) = 0, and lim(x→−2) s(x) = −3.

Step 4: Add the limits together: 4 + 0 + (-3).

Step 5: The result of the addition gives us the limit of the sum of the functions as x approaches -2.

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Key Concepts

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

Limits of Functions

A limit describes the value that a function approaches as the input approaches a certain point. In this case, we are interested in the behavior of the functions p(x), r(x), and s(x) as x approaches -2. Understanding limits is crucial for evaluating expressions involving functions at points where they may not be explicitly defined.

Limit Laws

Limit laws are rules that allow us to compute the limit of a sum, difference, product, or quotient of functions based on the limits of the individual functions. For example, the limit of a sum is the sum of the limits, provided the limits exist. This principle will be applied to find the limit of the expression p(x) + r(x) + s(x) as x approaches -2.

Evaluating Limits

Evaluating limits involves substituting the point of interest into the function, if possible, or applying limit laws to simplify the expression. In this problem, we will substitute the limits of p(x), r(x), and s(x) as x approaches -2 to find the overall limit of their sum. This process is essential for solving the given limit problem.