Lim x tends to 0 (Tanx - sinx …

Ans. \(\lim_{x \to 0} {tan x - sin x \over x^3}\)

=> \(\lim_{x \to 0} {{sin x\over cos x} - sin x \over x^3}\)

=> \(\lim_{x \to 0} {sin x({1\over cosx } - 1) \over x^3}\)

=> \(\lim_{x \to 0} {sin x(1 - cosx) \over x^3 . \space cos x }\)

=> \(\lim_{x \to 0} {[{1\over cos x}. {sin x \over x} .{ 1 - cosx \over x^2 }}]\)

=> \(\lim_{x \to 0} {[{1\over cos x}. {sin x \over x} .{ 2sin^2{x\over 2} \over x^2 }}]\)

=> \(2\lim_{x \to 0} {[{1\over cos x}. {sin x \over x} .{ sin{x\over 2} \over x}.{ sin{x\over 2} \over x}}]\)

=> \(2\lim_{x \to 0} {[{1\over cos x}. {sin x \over x} .{1\over 2}{ sin{x\over 2} \over {x\over 2}}.{1\over 2}{ sin{x\over 2} \over {x\over 2}}}]\)

=> \(2[\lim_{x \to 0} {{1\over cos x}\times \lim_{x \to 0}{sin x \over x} \times \lim_{x \to 0}{1\over 2}{ sin{x\over 2} \over {x\over 2}}\times \lim_{x \to 0}{1\over 2}{ sin{x\over 2} \over {x\over 2}}}]\)   

=> \(2[\lim_{x \to 0} {{1\over cos x}\times \lim_{x \to 0}{sin x \over x} \times \lim_{{x\over 2} \to 0}{1\over 2}{ sin{x\over 2} \over {x\over 2}}\times \lim_{{x\over2} \to 0}{1\over 2}{ sin{x\over 2} \over {x\over 2}}}]\)     [as x tends to zero, x/2 also tends to zero]

=> \(2\times 1 \times 1 \times {1\over 2}\times {1\over 2} = {1\over 2}\)