4. Applications of Derivatives
Differentials
Problem 4.R.116b
Textbook Question
Cosine limits Let n be a positive integer. Evaluate the following limits.
lim_x→0 (1 - cosⁿ x) / x²
Verified step by step guidance1
Recognize that the limit involves the expression (1 - cosⁿ x) as x approaches 0, which suggests using the Taylor series expansion for cos x.
Recall the Taylor series expansion for cos x around x = 0: cos x = 1 - (x²/2) + (x⁴/24) - ...
Substitute the Taylor series into the expression for cosⁿ x, noting that cosⁿ x can be approximated as (1 - (x²/2) + O(x⁴))ⁿ for small x.
Use the binomial expansion to simplify (1 - (x²/2) + O(x⁴))ⁿ, focusing on the leading term as x approaches 0.
Finally, substitute this simplified expression back into the limit and evaluate the limit as x approaches 0, considering the leading terms.
Was this helpful?
