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

lim x→1 x^2 − 1 / x − 1

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The given limit is \( \lim_{{x \to 1}} \frac{{x^2 - 1}}{{x - 1}} \). Notice that if you directly substitute \( x = 1 \), both the numerator and the denominator become zero, resulting in an indeterminate form \( \frac{0}{0} \).

The expression \( x^2 - 1 \) is a difference of squares, which can be factored as \( (x - 1)(x + 1) \).

Substitute the factored form into the limit: \( \lim_{{x \to 1}} \frac{{(x - 1)(x + 1)}}{{x - 1}} \).

The \( (x - 1) \) terms in the numerator and the denominator cancel each other out, simplifying the expression to \( \lim_{{x \to 1}} (x + 1) \).

Now that the expression is simplified, substitute \( x = 1 \) into \( x + 1 \) to find the limit.>

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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 specific points, which is crucial for evaluating functions that may not be defined at those points. In this case, we are interested in the limit as x approaches 1.

Factoring

Factoring is a mathematical process of breaking down an expression into simpler components, which can help simplify complex expressions. In the context of limits, factoring can be used to eliminate indeterminate forms, such as 0/0, by canceling common factors in the numerator and denominator. This technique is essential for evaluating the limit in the given problem.

Indeterminate Forms

Indeterminate forms occur when the limit of a function results in an ambiguous expression, such as 0/0 or ∞/∞. These forms require further analysis or manipulation to resolve. In the provided limit problem, substituting x = 1 directly leads to an indeterminate form, necessitating the use of factoring or other techniques to 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 x^2 − 1 / x − 1