Domanda del libro di testo
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 1) / (2x + 4) as
b. x→−2⁻
180
views
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 55d
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x²/2 − 1/x) as
d. x→−1
Guida verificata passo dopo passo1
Identify the function for which you need to find the limit: \( f(x) = \frac{x^2}{2} - \frac{1}{x} \).
Substitute \( x = -1 \) into the function to evaluate the limit directly: \( f(-1) = \frac{(-1)^2}{2} - \frac{1}{-1} \).
Calculate each term separately: \( \frac{(-1)^2}{2} = \frac{1}{2} \) and \( -\frac{1}{-1} = 1 \).
Combine the results from the previous step: \( \frac{1}{2} + 1 \).
Simplify the expression to find the limit as \( x \to -1 \).
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Limits
Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near specific points, including points of discontinuity or infinity. Evaluating limits is essential for determining the continuity of functions and for finding derivatives and integrals.
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05:50
One-Sided Limits
Rational Functions
A rational function is a function that can be expressed as the ratio of two polynomials. Understanding rational functions is crucial for limit evaluation, especially when determining behavior as the variable approaches a specific value. In this case, the function involves a polynomial in the numerator and a term involving x in the denominator, which can affect the limit as x approaches -1.
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6:04Intro to Rational Functions
Substitution in Limits
Substitution is a technique used in limit evaluation where you directly replace the variable with the value it approaches. This method is effective when the function is continuous at that point. However, if direct substitution leads to an indeterminate form, further analysis or algebraic manipulation may be necessary to find the limit.
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04:27Substitution With an Extra Variable
Pratica correlata
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Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x²/2 − 1/x) as
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Domanda del libro di testo
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x² − 3x + 2) / (x³ − 2x²) as
d. x→2
207
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