Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→−1 (2x − 1)^2 − 9 / x + 1

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Identify the limit expression: \( \lim_{{x \to -1}} \frac{{(2x - 1)^2 - 9}}{{x + 1}} \).

Notice that direct substitution of \( x = -1 \) results in an indeterminate form \( \frac{0}{0} \).

Factor the numerator: \((2x - 1)^2 - 9\) is a difference of squares, which can be factored as \((2x - 1 - 3)(2x - 1 + 3)\).

Simplify the expression: \((2x - 1 - 3)(2x - 1 + 3) = (2x - 4)(2x + 2)\).

Cancel the common factor \(x + 1\) from the numerator and denominator, then evaluate the limit by substituting \(x = -1\) into the simplified expression.

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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. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

Factoring and Simplifying Expressions

Factoring involves rewriting an expression as a product of its factors, which can simplify the evaluation of limits, especially when direct substitution leads to indeterminate forms like 0/0. In the given limit, factoring the numerator allows for cancellation with the denominator, making it easier to compute the limit as x approaches -1.

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. Recognizing these forms is essential, as they often require additional techniques, such as factoring, rationalizing, or applying L'Hôpital's Rule, to resolve and find the actual limit.

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.lim x→−1 (2x − 1)^2 − 9 / x + 1