hey i think its easy

lim  {(a-n)nx-tan x}/x   *   (sin nx)/x

lim  [ {(a-n)nx}/x  -(tanx/x)]  *  (sin nx/nx)*n

lim  {(a-n)n -1} * n        .........                {sinx/x=1 & tanx/x=1 as x-0}

now putting the limit=

a=n+1/n

hope u got it......