The value of limx →0 ( sin x / x)sinx /x -sinx is :

Dear Mainak,

Substituting the value x=0 in the limit we get this limit of the form

18 .Thus this is the indeterminate form of the limit.

We can write it as

lim x-->0 (1+(sin x - x)/x)^[sin x/(x-sin x)]

Making the use of the  property

 lim x-->0 (1+f(x))^(1/f(x))=e,when lim x-->0 f(x) =0

Therefore Rewriting above as

lim x-->0 (1+(sin x - x)/x)^[sin x/(x-sin x)][x/sin x]

or lim x-->0 (1+(sin x - x)/x)^[x/(sin x- x)][sin x/x](-1)

=     e^ lim x-->0 (sin x/x)(-1)

=e^(1)(-1)

=1/e

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