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

lim x→4 x^2 − 16 / 4 − x

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The expression \( \frac{x^2 - 16}{4 - x} \) is in an indeterminate form \( \frac{0}{0} \) as \( x \to 4 \).

Notice that \( x^2 - 16 \) is a difference of squares, which can be factored as \( (x - 4)(x + 4) \).

Substitute the factored form into the expression: \( \frac{(x - 4)(x + 4)}{4 - x} \).

Notice that \( 4 - x = -(x - 4) \), so the expression becomes \( \frac{(x - 4)(x + 4)}{-(x - 4)} \). Cancel \( x - 4 \) from the numerator and denominator, leaving \( -(x + 4) \).

Now that the expression is simplified to \( -(x + 4) \), substitute \( x = 4 \) 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 expressions that may be undefined at those points. In this case, we are interested in the limit as x approaches 4.

Factoring

Factoring is the process of breaking down an expression into simpler components, which can help simplify complex limits. In the given limit, the expression x^2 - 16 can be factored into (x - 4)(x + 4), allowing for cancellation of terms that may lead to an indeterminate form when directly substituting x = 4.

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 further manipulation, such as factoring or applying L'Hôpital's Rule, to evaluate the limit correctly.

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.lim x→4 x^2 − 16 / 4 − x