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

lim_x→0 (1 - cos xⁿ) / x²ⁿ

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Recognize that the limit involves an indeterminate form 0/0 as x approaches 0, which suggests the use of L'Hôpital's Rule or a series expansion.

Consider using the Taylor series expansion for cos(x) around x = 0: cos(x) ≈ 1 - x²/2 + x⁴/24 - ... . For cos(xⁿ), substitute xⁿ into the series: cos(xⁿ) ≈ 1 - (xⁿ)²/2 + (xⁿ)⁴/24 - ... .

Substitute the series expansion into the limit expression: (1 - (1 - (xⁿ)²/2 + ...)) / x²ⁿ = ((xⁿ)²/2 - ...) / x²ⁿ.

Simplify the expression: ((x²ⁿ)/2 - ...) / x²ⁿ = (1/2) - ... . As x approaches 0, higher order terms become negligible.

Conclude that the limit evaluates to 1/2 as x approaches 0, since the dominant term in the expansion is (1/2).

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Key Concepts

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

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 a fundamental concept used to define continuity, derivatives, and integrals. Evaluating limits often involves techniques such as substitution, factoring, or applying L'Hôpital's rule when dealing with indeterminate forms.

Taylor Series Expansion

The Taylor series expansion is a representation of 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 given by cos(x) = 1 - x²/2! + x⁴/4! - ... This series is useful for approximating functions and simplifying the evaluation of limits, especially when dealing with small values of x.

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 separately. This rule is particularly helpful in simplifying complex limit problems, such as the one presented in the question.

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