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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1:49 minutes

Problem 2.2.29

Textbook Question

Limits of quotients

Find the limits in Exercises 23–42.

limt→−2 (−2x − 4) / (x³ + 2x²)

Verified step by step guidance

Step 1: Identify the type of limit problem. This is a limit of a quotient as x approaches -2. We need to evaluate lim_{x→−2} (−2x − 4) / (x³ + 2x²).

Step 2: Check if direct substitution is possible. Substitute x = -2 into the numerator and denominator to see if it results in an indeterminate form like 0/0.

Step 3: Simplify the expression. Factor the denominator x³ + 2x² to see if it can be simplified. The denominator can be factored as x²(x + 2).

Step 4: Cancel common factors. If the numerator and denominator have common factors, cancel them to simplify the expression. Check if the numerator −2x − 4 can be factored to find common factors with the denominator.

Step 5: Evaluate the limit. After simplifying, substitute x = -2 again to find the limit. If the expression is still indeterminate, consider using L'Hôpital's Rule or further algebraic manipulation.

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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 help in understanding the behavior of functions near specific points, especially where they may not be defined. In this case, evaluating the limit as x approaches -2 will reveal the behavior of the quotient function near that point.

Quotient of Functions

The quotient of functions involves dividing one function by another. When finding limits of quotients, it is essential to consider the behavior of both the numerator and denominator as the variable approaches a specific value. If the denominator approaches zero, it may lead to undefined behavior or require further analysis, such as factoring or applying L'Hôpital's Rule.

Factoring and Simplifying

Factoring and simplifying expressions is a crucial step in limit problems, especially when dealing with quotients. By factoring the numerator and denominator, one can often cancel common terms, which can help eliminate indeterminate forms like 0/0. This simplification allows for easier evaluation of the limit as the variable approaches the specified value.