Limits of quotients

Find the limits in Exercises 23–42.

limx→−1 (√(x² + 8) − 3) / (x + 1)

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Recognize that the limit involves a quotient where direct substitution results in an indeterminate form 0/0. This suggests the need for algebraic manipulation.

Multiply the numerator and the denominator by the conjugate of the numerator, which is (√(x² + 8) + 3), to eliminate the square root in the numerator.

Simplify the expression by applying the difference of squares formula: (a - b)(a + b) = a² - b². Here, a = √(x² + 8) and b = 3.

After simplification, the numerator becomes (x² + 8) - 9, which simplifies further to x² - 1. Notice that x² - 1 can be factored as (x - 1)(x + 1).

Cancel the common factor (x + 1) from the numerator and the denominator, then evaluate the limit of the simplified expression as x approaches -1.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, we are interested in the limit of a quotient as x approaches -1. Understanding limits helps in analyzing the behavior of functions near points of interest, especially where they may not be directly computable.

Rational Functions

A rational function is a ratio of two polynomials. In the given limit problem, the expression involves a square root in the numerator and a linear polynomial in the denominator. Recognizing the structure of rational functions is crucial for applying limit laws and techniques, such as factoring or rationalizing, to simplify the expression before evaluating the limit.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. If direct substitution in the limit leads to such forms, L'Hôpital's Rule allows us to differentiate the numerator and denominator separately. This technique is particularly useful in the given problem, where direct substitution results in an indeterminate form, necessitating further analysis.

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