Domanda del libro di testo
Limits of quotients
Find the limits in Exercises 23–42.
limx→−1 (√(x² + 8) − 3) / (x + 1)
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Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 2, Problema 2.4.16
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
limh→0− (√6 − √(5h² + 11h + 6))/ h
Guida verificata passo dopo passo1
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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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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05:50
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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05:25Determine Continuity Algebraically
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.
Video consigliato:
6:04Intro to Rational Functions
Pratica correlata
Domanda del libro di testo
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
lim x→1 1/x = 1
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Domanda del libro di testo
At what points are the functions in Exercises 13–30 continuous?
y = 1/(x – 2) – 3x
235
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Domanda del libro di testo
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
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Domanda del libro di testo
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
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Domanda del libro di testo
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
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