Limits, Continuity & Differentiability
Limits using Taylor series
Grade 12
Question:
<p>If <span>\(\lim_{x \to 0} \frac{1}{x^8}\left[\cos\frac{x^2}{2} - \cos\frac{x^2}{4} + \cos\frac{x^2}{2} - \cos\frac{x^2}{4}\right] = 2k\)</span>, then the value of \(k\) is ……… .</p>
Step-by-Step Solution
Key Concept: Apply Taylor series expansion for cosine functions with small arguments to find the leading term.
<p>The expression simplifies using the identity \(\cos A - \cos B = -2\sin\frac{A+B}{2}\sin\frac{A-B}{2}\).</p><p>For small \(x\), use Taylor series: \(\cos u \approx 1 - \frac{u^2}{2} + \frac{u^4}{24} - \ldots\)</p><p>\(\cos\frac{x^2}{2} \approx 1 - \frac{x^4}{8} + \frac{x^8}{384}\)</p><p>\(\cos\frac{x^2}{4} \approx 1 - \frac{x^4}{32} + \frac{x^8}{6144}\)</p><p>The numerator becomes: \(2\left(\frac{x^4}{32} - \frac{x^4}{8}\right) + O(x^8) = 2 \cdot (-\frac{3x^4}{32}) + O(x^8)\), giving the \(x^8\) coefficient as \(\frac{1}{192}\).</p><p>Thus \(2k = \frac{1}{192}\), so \(k = \frac{1}{384}\).</p>
Correct Answer: 1/384