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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 2 - Limits문제 91
2장, 문제 91

Calculate the following limits using the factorization formula x^n−a^n=(x−a)(x^n−1+ax^n−2+a^2x^n−3+⋯+a^n−2x+a^n−1), where n is a positive integer and a is a real number.
lim x→1 x^6 − 1 / x − 1

검증된 단계별 안내
1
Recognize that the given limit is in the indeterminate form 0/0 when x approaches 1, so we need to simplify the expression.
Use the factorization formula for x^n - a^n: x^6 - 1 = (x - 1)(x^5 + x^4 + x^3 + x^2 + x + 1).
Substitute the factorized form into the limit: lim_{x \(\to\) 1} \(\frac{(x - 1)(x^5 + x^4 + x^3 + x^2 + x + 1)}{x - 1}\).
Cancel the common factor (x - 1) in the numerator and the denominator.
Evaluate the limit of the remaining expression x^5 + x^4 + x^3 + x^2 + x + 1 as x approaches 1.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 evaluating the limit of a rational function as x approaches 1, which often involves determining the behavior of the function near that point.
추천 영상:
05:50
One-Sided Limits

Factorization

Factorization is the process of breaking down an expression into a product of simpler factors. The provided factorization formula for x^n - a^n allows us to simplify the limit calculation by rewriting the expression in a form that eliminates indeterminate forms, such as 0/0, which can occur when directly substituting the limit point.
추천 영상:
06:11
Limits of Rational Functions: Denominator = 0

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. In this limit problem, x^6 - 1 is a polynomial, and understanding its structure helps in applying the factorization formula effectively to find the limit as x approaches 1.
추천 영상:
6:04
Introduction to Polynomial Functions
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