Question 1090295
{{{lim(x->0,(sin(2x))/(3x)) }}}

Move the term {{{1/3}}} outside of the limit because it is constant with respect to {{{x}}}.

{{{(1/3)lim(x->0,(sin(2x))/x) }}}

Evaluate the limit of the numerator and the limit of the denominator:
you got
{{{0/0 }}}

Since {{{0/0}}} is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

{{{(1/3)lim(x->0,(sin(2x)/x))=lim(x->0,((d/dx)*sin(2x))/((d/dx)x))}}}


{{{(1/3)lim(x->0,2cos(2x)/1)}}}

{{{(1/3)*2cos(2lim(x->0,x))}}}

{{{(1/3)*2cos(2*0)}}}

{{{(1/3)*2cos(0)}}}

{{{(1/3)*2*1}}}

{{{2/3}}}


so, {{{lim(x->0,(sin(2x))/(3x))=2/3 }}}