Cosine limits Let n be a positive integer. Evaluate the following limits.

lim_x→0 (1 - cosⁿ x) / x²

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Recognize that the limit involves a trigonometric function, specifically cosine, raised to the power of n. This suggests that we might need to use a trigonometric identity or series expansion to simplify the expression.

Recall the Taylor series expansion for cos(x) around x = 0: cos(x) ≈ 1 - x²/2 + x⁴/24 - ... . For small values of x, higher-order terms become negligible.

Substitute the Taylor series expansion of cos(x) into the expression 1 - cosⁿ(x). For small x, cosⁿ(x) can be approximated as (1 - x²/2)ⁿ.

Use the binomial expansion for (1 - x²/2)ⁿ to approximate it as 1 - n(x²/2) + higher-order terms. This simplifies the expression 1 - cosⁿ(x) to n(x²/2) for small x.

Substitute this approximation into the limit expression: lim_x→0 (1 - cosⁿ(x)) / x² ≈ lim_x→0 (n(x²/2)) / x². Simplify this expression to find the limit as x approaches 0.

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Limit of a Function

In calculus, the limit of a function describes the behavior of that function as its input approaches a certain value. It is essential for evaluating expressions that may be indeterminate at specific points, such as when both the numerator and denominator approach zero. Understanding limits allows us to analyze the continuity and behavior of functions near points of interest.

Taylor Series Expansion

The Taylor series expansion is a powerful tool in calculus that expresses a function as an infinite sum of terms calculated from the values of its derivatives at a single point. For the cosine function, the expansion around zero is particularly useful, as it provides a polynomial approximation that simplifies the evaluation of limits. This concept is crucial for approximating functions and understanding their behavior near specific points.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in calculus for simplifying complex limit problems, especially when dealing with trigonometric functions.

Cosine limits Let n be a positive integer. Evaluate the following limits.lim_x→0 (1 - cosⁿ x) / x²