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

lim x→1 5x^2 + 6x + 1 / 8x − 4

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Identify the limit expression: \(\lim_{{x \to 1}} \frac{5x^2 + 6x + 1}{8x - 4}\).

Substitute \(x = 1\) into the expression to check if it results in an indeterminate form: \(\frac{5(1)^2 + 6(1) + 1}{8(1) - 4}\).

Calculate the numerator and denominator separately: Numerator: \(5(1)^2 + 6(1) + 1 = 12\), Denominator: \(8(1) - 4 = 4\).

Since the substitution does not result in an indeterminate form, evaluate the limit by simplifying the expression: \(\frac{12}{4}\).

Conclude the evaluation by simplifying the fraction 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 directly computable at those points. Limits can be finite or infinite and are essential for defining derivatives and integrals.

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Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given limit problem, the expression involves a rational function where the numerator and denominator are both polynomials. Understanding how to simplify and analyze rational functions is key to finding limits, especially when dealing with points that may lead to indeterminate forms.

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Intro to Rational Functions

Indeterminate Forms

Indeterminate forms occur when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In such cases, techniques such as factoring, rationalizing, or applying L'Hôpital's Rule may be necessary to resolve the limit. Recognizing these forms is crucial for correctly determining the limit of a function.

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