Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (x² − x + sin x) / 2x

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First, identify the limit expression: lim(x→0) (x² − x + sin x) / 2x.

Break down the expression into separate terms: (x²/2x), (-x/2x), and (sin x/2x).

Simplify each term individually: (x²/2x) simplifies to x/2, (-x/2x) simplifies to -1/2, and (sin x/2x) can be approached using the limit property lim(θ→0) sin θ/θ = 1.

Apply the limit property to the term (sin x/2x): as x approaches 0, sin x/x approaches 1, so sin x/2x approaches 1/2.

Combine the simplified terms and apply the limit: lim(x→0) [x/2 - 1/2 + 1/2]. Evaluate each term as x approaches 0 to find the overall limit.

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Limit of a Function

The limit of a function describes the value that a function approaches as the input approaches a certain point. In calculus, limits are fundamental for understanding continuity, derivatives, and integrals. The notation lim x→a f(x) indicates the limit of f(x) as x approaches a. Evaluating limits often involves techniques such as substitution, factoring, or applying special limit results like L'Hôpital's Rule.

Sine Function and Its Limit

The limit lim θ→0 sin θ / θ = 1 is a crucial result in calculus, particularly in the study of trigonometric functions. This limit shows that as θ approaches 0, the ratio of sin θ to θ approaches 1, which is essential for deriving derivatives of sine and cosine functions. This result is often used in evaluating limits involving sine functions and is foundational for understanding the behavior of trigonometric functions near zero.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits of indeterminate forms, such as 0/0 or ∞/∞. When faced with such forms, the rule states that the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This process can be repeated if the limit remains indeterminate, making it a powerful tool for solving complex limit problems in calculus.

Using limθ→0 sin θ / θ = 1Find the limits in Exercises 23–46.limx→0 (x² − x + sin x) / 2x