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Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 θcos θ
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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 2.2.43
Limits with trigonometric functions
Find the limits in Exercises 43–50.
limx→0 (2sin x − 1)
Guida verificata passo dopo passo1
Identify the limit expression: \( \lim_{x \to 0} (2\sin x - 1) \).
Recall the basic limit property for sine: \( \lim_{x \to 0} \sin x = 0 \).
Substitute the limit of \( \sin x \) into the expression: \( 2(\lim_{x \to 0} \sin x) - 1 \).
Simplify the expression using the known limit: \( 2(0) - 1 \).
Conclude the limit evaluation by simplifying the expression to find the result.
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Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, we are interested in the limit of the function as x approaches 0, which helps determine the behavior of the function near that point.
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05:50
One-Sided Limits
Trigonometric Functions
Trigonometric functions, such as sine and cosine, relate angles to ratios of sides in right triangles. They are periodic functions and play a crucial role in calculus, especially when evaluating limits involving angles. The sine function, in particular, has specific limit properties that are useful when x approaches 0, such as sin(x) approaching x.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Sine Limit Property
The sine limit property states that as x approaches 0, the limit of sin(x)/x equals 1. This property is pivotal when evaluating limits involving sine functions, as it allows simplification of expressions. In the given limit, understanding this property will help in determining the limit of the expression 2sin(x) as x approaches 0.
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06:21Properties of Functions
Pratica correlata
Domanda del libro di testo
Using the Sandwich Theorem
If √(5 −2x²) ≤ f(x) ≤ √(5−x²) for −1 ≤ x ≤ 1, find limx→0 f(x).
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Existence of Limits
In Exercises 5 and 6, explain why the limits do not exist.
limx→0 x/|x|
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Slope of a Curve at a Point
In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.
y=x³, P(2,8)
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Domanda del libro di testo
At what points are the functions in Exercises 13–30 continuous?
y = 1/(|x| + 1) − x²/2
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Domanda del libro di testo
Limits of quotients
Find the limits in Exercises 23–42.
limt→−2 (−2x − 4) / (x³ + 2x²)
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