Textbook Question
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx →4 (9 − x) = 5
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 2.4.16
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
limh→0− (√6 − √(5h² + 11h + 6))/ h
Verified step by step guidance1
Identify the expression for the one-sided limit: limh→0− (√6 − √(5h² + 11h + 6))/h. This is a left-hand limit as h approaches 0 from the negative side.
Recognize that direct substitution of h = 0 results in an indeterminate form 0/0. To resolve this, we can use algebraic manipulation, such as multiplying by the conjugate.
Multiply the numerator and the denominator by the conjugate of the numerator: (√6 + √(5h² + 11h + 6)). This will help eliminate the square roots in the numerator.
Simplify the expression: The numerator becomes (6 - (5h² + 11h + 6)) = -5h² - 11h. The denominator becomes h(√6 + √(5h² + 11h + 6)).
Factor out h from the numerator: -h(5h + 11). Cancel the h in the numerator and denominator, then evaluate the limit as h approaches 0 from the left.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
One-Sided Limits
One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side, either the left (denoted as h→0−) or the right (h→0+). Understanding one-sided limits is crucial for analyzing functions that may behave differently from each side of a point, especially at points of discontinuity or where the function is not defined.
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One-Sided Limits
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions to make limits easier to evaluate. In the context of limits, this often includes factoring, rationalizing, or combining terms to eliminate indeterminate forms like 0/0, which can arise when directly substituting the limit value into the function.
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Rationalization
Rationalization is a technique used to eliminate square roots or other irrational expressions from the denominator or numerator of a fraction. In the given limit problem, rationalizing the numerator can help simplify the expression, making it easier to evaluate the limit as h approaches 0 from the left.
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6:04Intro to Rational Functions
Related Practice
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 = -2
Textbook Question
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limx→0 (1 − cos 3x) / 2x
Textbook Question
Finding Limits of Differences When x → ±∞
Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)
lim x → ∞ (√(9x² − x) − 3x)
Textbook Question
Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?
lim ϴ → 0 cos (πϴ/sin ϴ)
Textbook Question
Theory and Examples
If limx→4 (f(x) − 5) / (x − 2) = 1, find limx→4 f(x).