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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3:17 minutes

Problem 2.6.33

Textbook Question

Evaluate each limit and justify your answer.
lim x→4 √x^3−2x^2−8x / x−4

Verified step by step guidance

Identify the type of limit: This is a limit of the form \( \frac{0}{0} \) as both the numerator and denominator approach zero when \( x \to 4 \).

Apply L'Hôpital's Rule: Since the limit is of the indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule, which involves differentiating the numerator and the denominator separately.

Differentiate the numerator: Find the derivative of \( \sqrt{x^3 - 2x^2 - 8x} \) with respect to \( x \). Use the chain rule and power rule for differentiation.

Differentiate the denominator: The derivative of \( x - 4 \) with respect to \( x \) is simply 1.

Evaluate the new limit: Substitute \( x = 4 \) into the new expression obtained after applying L'Hôpital's Rule and simplify to find the limit.

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Key Concepts

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

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. Evaluating limits often involves techniques such as substitution, factoring, or applying L'Hôpital's rule when dealing with indeterminate forms.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In such cases, further analysis is required, often involving algebraic manipulation or L'Hôpital's rule, which allows for differentiation of the numerator and denominator to resolve the limit.

Factoring and Simplification

Factoring and simplification are techniques used to rewrite expressions in a more manageable form, especially when evaluating limits. By factoring out common terms or simplifying complex expressions, one can often eliminate indeterminate forms and make it easier to compute the limit. This process is crucial in finding the limit of rational functions or polynomials.

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