Determine the following limits.
lim x→3 x^4 − 81 / x − 3

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Identify the form of the limit: Substitute \( x = 3 \) into the expression \( \frac{x^4 - 81}{x - 3} \) to see if it results in an indeterminate form like \( \frac{0}{0} \).

Recognize that \( x^4 - 81 \) can be factored as a difference of squares: \( x^4 - 81 = (x^2 - 9)(x^2 + 9) \).

Further factor \( x^2 - 9 \) as a difference of squares: \( x^2 - 9 = (x - 3)(x + 3) \).

Rewrite the original expression using the factored form: \( \frac{(x - 3)(x + 3)(x^2 + 9)}{x - 3} \).

Cancel the common factor \( x - 3 \) in the numerator and denominator, then evaluate the limit of the simplified expression as \( x \to 3 \).

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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 x approaches 3, which requires evaluating the function's behavior close to that point.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. In the given limit, the expression x^4 - 81 can be factored using the difference of squares, which simplifies the limit calculation by eliminating the indeterminate form that occurs when directly substituting x = 3.

Indeterminate Forms

Indeterminate forms arise in calculus when direct substitution in a limit leads to an undefined expression, such as 0/0 or ∞/∞. In this problem, substituting x = 3 into the limit results in the form 0/0, indicating that further analysis, such as factoring or applying L'Hôpital's Rule, is necessary to resolve the limit and find a meaningful value.

Determine the following limits.lim x→3 x^4 − 81 / x − 3