lim f(x)=(sinx-x)/(x^3), where x-->0 Solution follow this Let lim ( x -> 0 ) ( x
ID: 3283923 • Letter: L
Question
lim f(x)=(sinx-x)/(x^3), where x-->0Explanation / Answer
follow this Let lim ( x -> 0 ) ( x - sin x ) / x^3 = L ............(1) ......................................… In (1), put x = 3y so that , as x -> 0, y -> 0. ......................................… Hence, the limit L in (1) now becomes L = lim ( y -> 0 ) [ ( 3y ) - sin ( 3y ) ] / ( 3y )^3 = lim [ 3y - ( 3 sin y - 4 sin^3 y ) ] / ( 27 y^3 ) = ( 1/27 ) lim [ 3( y - sin y ) + 4 sin^3 y ] / y^3 = ( 1/27 ) { (3) lim [ ( y - sin y ) / y^3 ] + 4 lim ( sin y / y )^3 } = (1/27) { 3( L ) + 4 ( 1 ) }...............from (1) = ( L / 9 ) + ( 4 / 27 ) ......................................… Hence, L = ( L/9) + ( 4/27) 27 L = 3 L + 4 24 L = 4 L = 1 / 6. .......................Ans.
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