Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→4 1/x−1/4 / x − 4

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Recognize that the given limit is in the indeterminate form \( \frac{0}{0} \) as \( x \to 4 \).

Apply L'Hôpital's Rule, which states that if the limit \( \lim_{x \to c} \frac{f(x)}{g(x)} \) is in the form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \), provided the latter limit exists.

Differentiate the numerator \( f(x) = \frac{1}{x} - \frac{1}{4} \) to get \( f'(x) = -\frac{1}{x^2} \).

Differentiate the denominator \( g(x) = x - 4 \) to get \( g'(x) = 1 \).

Evaluate the new limit \( \lim_{x \to 4} \frac{-\frac{1}{x^2}}{1} \) by substituting \( x = 4 \) into the 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. In this question, we are tasked with finding the limit of a function as x approaches 4.

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One-Sided Limits

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In the given limit, substituting x = 4 results in the form 0/0, indicating that further analysis, such as algebraic manipulation or L'Hôpital's Rule, is necessary to resolve the limit.

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L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms. It states that if the limit of f(x)/g(x) results in 0/0 or ∞/∞, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule simplifies the process of finding limits and is particularly useful in cases like the one presented in the question.

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