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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.2.33
Chapter 2, Problem 2.2.33

Limits of quotients


Find the limits in Exercises 23–42.


limu→1 (u⁴ − 1)/(u³ − 1)

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1
Identify the limit expression: \( \lim_{{u \to 1}} \frac{{u^4 - 1}}{{u^3 - 1}} \). Notice that direct substitution of \( u = 1 \) results in an indeterminate form \( \frac{0}{0} \).
Factor both the numerator and the denominator. The numerator \( u^4 - 1 \) can be factored as \( (u^2 + 1)(u - 1)(u + 1) \) using the difference of squares. The denominator \( u^3 - 1 \) can be factored as \( (u - 1)(u^2 + u + 1) \) using the difference of cubes.
Rewrite the limit expression using the factored forms: \( \lim_{{u \to 1}} \frac{{(u^2 + 1)(u - 1)(u + 1)}}{{(u - 1)(u^2 + u + 1)}} \).
Cancel the common factor \( (u - 1) \) from the numerator and the denominator, simplifying the expression to \( \lim_{{u \to 1}} \frac{{(u^2 + 1)(u + 1)}}{{u^2 + u + 1}} \).
Substitute \( u = 1 \) into the simplified expression to find the limit: \( \frac{{(1^2 + 1)(1 + 1)}}{{1^2 + 1 + 1}} \). Calculate the expression to determine the limit.

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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. They help in understanding the behavior of functions near specific points, especially when direct substitution may lead to indeterminate forms. In this case, evaluating the limit as u approaches 1 requires careful analysis of the function's behavior around that point.
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One-Sided Limits

Quotient of Functions

The quotient of functions involves dividing one function by another, which can introduce complexities, especially when the denominator approaches zero. In the limit problem presented, the expression (u⁴ - 1)/(u³ - 1) is a quotient, and understanding how to simplify or manipulate this expression is crucial for finding the limit. Techniques such as factoring or applying L'Hôpital's Rule may be necessary.
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The Quotient Rule

Factoring Polynomials

Factoring polynomials is a technique used to simplify expressions, particularly when evaluating limits. In the given limit, both the numerator and denominator can be factored to identify common terms that may cancel out, allowing for a clearer evaluation of the limit. Recognizing patterns in polynomial expressions, such as the difference of squares or cubes, is essential for effective simplification.
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Introduction to Polynomial Functions
Related Practice
Textbook Question

A function value Show that the function F(x) = ( x − a)²(x − b)² + x takes on the value (a + b)² for some value of x.

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Textbook Question

Formal Definitions of One-Sided Limits


Greatest integer function Find (a) limx→400+ ⌊x⌋ and (b) limx→400− ⌊x⌋; then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about limx→400 ⌊x⌋? Give reasons for your answer.

Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limx→0 (x² − x + sin x) / 2x

Textbook Question

At what points are the functions in Exercises 13–30 continuous?

y = √(x⁴ +1)/(1 + sin² x)

Textbook Question

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx→−3 (x² − 9) / (x + 3) = −6

Textbook Question

Suppose that a function f(x) is defined for all x in [-1,1]. Can anything be said about the existence of limx→0 f(x)? Give reasons for your answer.