4. Applications of Derivatives
Differentials
Problem 4.R.116a
Textbook Question
Cosine limits Let n be a positive integer. Evaluate the following limits.
lim_x→0 (1 - cos xⁿ) / x²ⁿ
Verified step by step guidance1
Recognize that the limit involves the expression (1 - cos(xⁿ)) as x approaches 0, which suggests using the Taylor series expansion for cos(x) around x = 0.
Recall the Taylor series expansion for cos(x): cos(x) = 1 - (x²/2) + (x⁴/24) - ... . Therefore, for small values of x, we can approximate cos(xⁿ) as cos(xⁿ) ≈ 1 - (xⁿ)²/2.
Substitute the approximation into the limit: lim_{x→0} (1 - (1 - (xⁿ)²/2)) / x²ⁿ, which simplifies to lim_{x→0} ((xⁿ)²/2) / x²ⁿ.
Simplify the expression further: ((xⁿ)²/2) / x²ⁿ = (x²ⁿ/2) / x²ⁿ = 1/2, since the x²ⁿ terms cancel out.
Evaluate the limit as x approaches 0, which gives the final result of 1/2.
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