Textbook Question
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
limh→0− (√6 − √(5h² + 11h + 6))/ h
265
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Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 2, Problem 2.2.38
Limits of quotients
Find the limits in Exercises 23–42.
limx→−1 (√(x² + 8) − 3) / (x + 1)
Verified step by step guidance1
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.
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One-Sided Limits
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.
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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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Guided course
5:50Power Rules
Related Practice
Textbook Question
Limits and Infinity
Find the limits in Exercises 37–46.
x⁴ + x³
lim -----------------
x→∞ 12x³ + 128
303
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Textbook Question
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
lim x→1 1/x = 1
255
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Textbook Question
Finding Limits
In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)
f(x) = 2/x − 3
314
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Textbook Question
At what points are the functions in Exercises 13–30 continuous?
f(x) = { (x³ − 8)/(x² − 4), x ≠ 2, x ≠ −2
3, x = 2
4, x = −2
209
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Textbook Question
Limits of Average Rates of Change
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.
f(x) = x², x = 1
333
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