Domanda del libro di testo
Infinite Limits
Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.
lim x→0⁺ 1 / 3x
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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 2.2.17
Calculating Limits
Find the limits in Exercises 11–22.
limx→−1/2 4x(3x+4)²
Guida verificata passo dopo passo1
Identify the limit expression: \( \lim_{x \to -\frac{1}{2}} 4x(3x+4)^2 \).
Substitute \( x = -\frac{1}{2} \) directly into the expression to check if it results in a determinate form.
Calculate \( 3x + 4 \) by substituting \( x = -\frac{1}{2} \), which gives \( 3(-\frac{1}{2}) + 4 = -\frac{3}{2} + 4 = \frac{5}{2} \).
Substitute \( x = -\frac{1}{2} \) into \( 4x \), which gives \( 4(-\frac{1}{2}) = -2 \).
Combine the results: \( -2 \times (\frac{5}{2})^2 \) to find the limit. Simplify the expression to complete the calculation.
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Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for defining derivatives and integrals. In this case, we are interested in the limit of the function as x approaches -1/2.
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One-Sided Limits
Polynomial Functions
Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function in the limit problem, 4x(3x+4)², is a polynomial function, and understanding its structure is essential for evaluating limits. Polynomial functions are continuous everywhere, which simplifies the process of finding limits.
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6:04Introduction to Polynomial Functions
Substitution Method
The substitution method is a technique used to evaluate limits by directly substituting the value that x approaches into the function, provided the function is continuous at that point. If direct substitution results in an indeterminate form, further algebraic manipulation may be necessary. In this case, substituting x = -1/2 into the polynomial will help find the limit.
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04:27Substitution With an Extra Variable
Pratica correlata
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) = 1/x, x = -2
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Domanda del libro di testo
Finding Limits of Differences When x → ±∞
Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)
lim x → ∞ (√(x + 9) − √(x + 4))
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Domanda del libro di testo
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx→1 f(x) = 1 if f(x) = {x², x ≠ 1
2, x = 1
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Domanda del libro di testo
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
lim x→0 x² sin (1/x) = 0
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Domanda del libro di testo
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limx→0 (1 − cos 3x) / 2x
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