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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2:12 minutes

Problem 22

Textbook Question

Determine the following limits.
lim p→1 p^5 − 1 / p − 1

Verified step by step guidance

Recognize that the limit \( \lim_{p \to 1} \frac{p^5 - 1}{p - 1} \) is an indeterminate form \( \frac{0}{0} \), which suggests using algebraic manipulation or L'Hôpital's Rule.

Factor the numerator \( p^5 - 1 \) using the difference of powers formula: \( p^5 - 1 = (p - 1)(p^4 + p^3 + p^2 + p + 1) \).

Substitute the factored form into the limit: \( \lim_{p \to 1} \frac{(p - 1)(p^4 + p^3 + p^2 + p + 1)}{p - 1} \).

Cancel the common factor \( p - 1 \) in the numerator and denominator: \( \lim_{p \to 1} (p^4 + p^3 + p^2 + p + 1) \).

Evaluate the limit by direct substitution: substitute \( p = 1 \) into the simplified expression \( p^4 + p^3 + p^2 + p + 1 \).

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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. In this case, we are interested in the limit as p approaches 1, which requires evaluating the function's behavior close to that point.

Factoring

Factoring is a mathematical process of breaking down an expression into simpler components, or factors, that can be multiplied together to yield the original expression. In the context of limits, factoring can help simplify expressions that yield indeterminate forms, such as 0/0. For the given limit, factoring the numerator p^5 - 1 can help resolve the limit as p approaches 1.

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to ambiguous results, such as 0/0 or ∞/∞. These forms require further analysis or manipulation to resolve. In the limit provided, substituting p = 1 directly results in the indeterminate form 0/0, necessitating techniques like factoring or L'Hôpital's rule to find the actual limit.

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