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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.2.43
Chapter 2, Problem 2.2.43

Limits with trigonometric functions


Find the limits in Exercises 43–50.


limx→0 (2sin x − 1)

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1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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Introduction 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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Properties of Functions
Related Practice
Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limθ→0 θcos θ

Textbook Question

Using the Sandwich Theorem


If √(5 −2x²) ≤ f(x) ≤ √(5−x²) for −1 ≤ x ≤ 1, find limx→0 f(x).

Textbook Question

Existence of Limits


In Exercises 5 and 6, explain why the limits do not exist.


limx→0 x/|x|

Textbook Question

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)

Textbook Question

Domains and Asymptotes


Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.


y = x³ / (x³ − 8)

Textbook Question

At what points are the functions in Exercises 13–30 continuous?

y = 1/(|x| + 1) − x²/2