Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 2 - Limits

Problem 91

Chapter 2, Problem 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

Verified step by step guidance

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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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 evaluating the limit of a rational function as x approaches 1, which often involves determining the behavior of the function near that point.

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.

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.

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Related Practice

Textbook Question

Let g(x)= {1 if x≥0

−1 if x<0.

a. Write a formula for |g(x)|.

Textbook Question

Suppose you park your car at a trailhead in a national park and begin a 2-hr hike to a lake at 7 A.M. on a Friday morning. On Sunday morning, you leave the lake at 7 A.M. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let f(t) be your distance from the car t hours after 7 a.m. on Friday morning, and let g(t) be your distance from the car t hours after 7 a.m. on Sunday morning.

b. Let h(t)=f(t)−g(t). Find h(0) and h(2).

Textbook Question

Use an appropriate limit definition to prove the following limits.

lim x→1 (5x−2) =3;

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Textbook Question

Suppose f(x) = {x^2 − 5x + 6 / x − 3 if x≠3

a if x=3.

Determine a value of the constant a for which lim x→3 f(x) = f(3).

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Textbook Question

a. Evaluate f(0), f(2), g(0), and g(2).