Question:

<div class="solution"><p><strong>LHD of e^{-|x|} at x=0:</strong> For x<0, e^{-|x|}=eˣ. LHD = eˣ|₀ = 1... wait: LHD = lim(h→0⁻)(e^h - 1)/h = 1. But |x| for x<0 means e^{-(-x)}=eˣ, derivative = eˣ|₀ = 1. Actually the left-hand derivative of e^{-|x|} = derivative of eˣ at x=0 = 1.</p><p><strong>Evaluate limit:</strong> <span class="math-inline">\(1-\cos(x/2)\approx x^2/8\)</span> for small x. <span class="math-inline">\(1-\cos(x^2/8)\approx x^4/128\)</span>. So limit = <span class="math-inline">\(\lim_{x\to 0}\frac{x^4/128}{2^m x^n}=\frac{1}{128\cdot 2^m}\cdot x^{4-n}\)</span>. For finite nonzero limit: n=4. Value = <span class="math-inline">\(1/(128\cdot 2^m)=1\implies 2^m=1/128=2^{-7}\implies m=-7\)</span>. Then |n+2m|=|4-14|=10. Divisible by 2 and 5.</p><p><strong>Answer: (A) 2 and (C) 5</strong></p><div class="key-concept"><strong>Key Concept:</strong> Nested cosine expansion: 1-cos(1-cos θ) ≈ θ⁴/8 for small θ</div></div>