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

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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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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.

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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.

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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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