सिद्ध कीजिये कि ` lim _ ( x to 0) ( tan x - sin x)/(x^(3))=1/2`

Explanation

बाया पक्ष =` lim _( xto 0) ( tan x- sin x)/(x^(3))`
` = lim _ ( x to0) ( sin x- sin x . cos x)/( x^(3) . cos x)`
` = lim _( x to 0) ( sin x.( 1- cos x))/( x^(3) . cos x)`
` = lim _ ( x to 0) ( sin x.2 sin^(2). 1/2 x)/( x^(3).cos x)`
` = lim_ ( xto0) ( sinx)/x .2/(cos x) ((sin. 1/2x)/(1/2x))^(2) .1/2^(2)`
` = lim_ ( xto0) ( sin x)/x xx lim_( xto0) 1/( cos x) xx lim _ ( x to0) ((sin. 1/2x)/(1/2x))^(2)xx1/2`
` = 1 xx 1 xx lim _(x to 0) ((sin. 1/2x)/(1/2x))^(2) xx 1/2 " " ( :' ( sin theta)/theta = 1 , " जब " theta to 0)`
` = 1/2 =` दाया पक्ष | इति सिद्धम |