Question

The value of \[ \lim_{x \to 0} \frac{1 - \cos(1 - \cos x)}{x^4} \] is:

• A. \( \frac{1}{6} \)
• B. \( \frac{1}{8} \)
• C. \( \frac{1}{10} \)
• D. \( \frac{1}{12} \)

Hint:

For limits involving trigonometric functions, use Taylor series expansions around the point of interest (in this case, \( x = 0 \)) to simplify the expressions and evaluate the limit.

Answer 1

The Correct Option is B

We are given the limit expression: \[ \lim_{x \to 0} \frac{1 - \cos(1 - \cos x)}{x^4} \] To solve this, we first expand \( \cos x \) around \( x = 0 \) using the Taylor series: \[ \cos x = 1 - \frac{x^2}{2} + O(x^4) \] Substitute this expansion into the expression \( 1 - \cos x \): \[ 1 - \cos x = \frac{x^2}{2} + O(x^4) \] Now, substitute this into \( 1 - \cos(1 - \cos x) \), and expand the cosine term similarly: \[ 1 - \cos(1 - \cos x) = 1 - \cos\left(\frac{x^2}{2} + O(x^4)\right) \] Using the Taylor expansion for cosine again: \[ \cos\left(\frac{x^2}{2} + O(x^4)\right) = 1 - \frac{1}{2} \left(\frac{x^2}{2} + O(x^4)\right)^2 \] Simplifying: \[ 1 - \cos(1 - \cos x) = \frac{x^4}{8} + O(x^6) \] Now, substitute this into the original limit expression: \[ \lim_{x \to 0} \frac{\frac{x^4}{8} + O(x^6)}{x^4} \] This simplifies to: \[ \frac{1}{8} \] Thus, the value of the limit is \( \frac{1}{8} \), making the correct answer Option B.