Medium Practice — Set 1

Quick Recap — Limit Techniques (JEE Application)

• $1^\infty$: $\displaystyle\lim f^{g}=e^{\lim g(f-1)}$; $\tfrac00$: factor/rationalise, standard limits, or Taylor series.
• Series: $\sin x=x-\tfrac{x^3}{6}+\dots$, $\cos x=1-\tfrac{x^2}{2}+\tfrac{x^4}{24}-\dots$, $\tan x=x+\tfrac{x^3}{3}+\dots$, $e^x=1+x+\tfrac{x^2}{2}+\dots$, $\ln(1+x)=x-\tfrac{x^2}{2}+\dots$.
• $1-\cos(kx)\approx\tfrac{k^2x^2}{2}$; for a product, $1-\prod\cos(k_ix)\approx\tfrac{x^2}{2}\sum k_i^2$.
• For $x\to\infty$ rational-power forms, pull out the dominant term before taking the limit.