Hard Practice — Advanced Set

Quick Recap — Multi-concept Limits & Differentiability (JEE Main level)

Chain the tools: higher-order Taylor expansions, the $1^\infty$ exponential trick, and simultaneous value + slope matching for piecewise functions. Many are numerical-value (Section B).

• $\sin x=x-\tfrac{x^3}{6}+\tfrac{x^5}{120}-\dots$, $\cos x=1-\tfrac{x^2}{2}+\tfrac{x^4}{24}-\dots$, $\tan x=x+\tfrac{x^3}{3}+\dots$, $\tan^{-1}x=x-\tfrac{x^3}{3}+\dots$, $\sin^{-1}x=x+\tfrac{x^3}{6}+\dots$.
• $1-\prod_{i}\cos(k_i x)\approx\tfrac{x^2}{2}\sum_i k_i^2$; differentiability of $|f|$ fails where $f$ changes sign.