Domanda del libro di testo
Graphing Simple Rational Functions
Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.
y = −3/(x − 3)
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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 2.6.90
Finding Limits of Differences When x → ±∞
Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)
lim x → ∞ (√(x² + x) − √(x² − x))
Guida verificata passo dopo passo1
Identify the expression whose limit you need to find: \( \lim_{x \to \infty} (\sqrt{x^2 + x} - \sqrt{x^2 - x}) \).
To simplify the expression, multiply and divide by the conjugate: \( \frac{(\sqrt{x^2 + x} - \sqrt{x^2 - x})(\sqrt{x^2 + x} + \sqrt{x^2 - x})}{\sqrt{x^2 + x} + \sqrt{x^2 - x}} \).
The numerator becomes a difference of squares: \((x^2 + x) - (x^2 - x) = 2x\).
The expression simplifies to \( \frac{2x}{\sqrt{x^2 + x} + \sqrt{x^2 - x}} \).
Divide the numerator and the denominator by \(x\) to simplify further: \( \frac{2}{\sqrt{1 + \frac{1}{x}} + \sqrt{1 - \frac{1}{x}}} \), and evaluate the limit as \(x \to \infty\).
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Limits at Infinity
Limits at infinity involve finding the value that a function approaches as the input grows indefinitely large or small. This concept is crucial for understanding the behavior of functions as x approaches positive or negative infinity, often simplifying complex expressions to determine their end behavior.
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03:07
Cases Where Limits Do Not Exist
Conjugate Multiplication
Multiplying by the conjugate is a technique used to simplify expressions, especially those involving square roots. By multiplying the numerator and denominator by the conjugate, we can eliminate radicals, making it easier to evaluate limits and simplify expressions.
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06:13Limits of Rational Functions with Radicals
Simplifying Radical Expressions
Simplifying radical expressions involves manipulating terms to reduce complexity, often by rationalizing denominators or combining like terms. This process is essential for evaluating limits, as it allows for clearer insight into the behavior of functions involving square roots or other radicals.
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6:36Simplifying Trig Expressions
Pratica correlata
Domanda del libro di testo
Limits with trigonometric functions
Find the limits in Exercises 43–50.
limx→0 (1 + x + sin x) / (3 cosx)
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Domanda del libro di testo
Exercises 5–10 refer to the function
f(x) = { x² − 1, −1 ≤ x < 0
2x, 0 < x < 1
1, x = 1
−2x + 4, 1 < x < 2
0, 2 < x < 3
graphed in the accompanying figure.
<IMAGE>
At what values of x is f continuous?
222
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Domanda del libro di testo
Use formal definitions to prove the limit statements in Exercises 93–96.
lim x → 0 (1 / |x|) = ∞
380
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Domanda del libro di testo
Graphing Simple Rational Functions
Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.
y = 1/(x − 1)
348
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Domanda del libro di testo
Using the Formal Definitions
Use the formal definitions of limits as x → ±∞ to establish the limits in Exercises 91 and 92.
If f has the constant value f(x) = k, then lim x → ∞ f(x) = k.
315
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