Limit

Yes lim is 3.     Use L'Hopital's theorem.  That is   lim  f(x)/ g(x)      =  lim f'(x) /g'x)   and you will get

lim  =ln'(1+3x)  /(x')   =    [ {1/(1+3x)} (3)  ]  /1   = 3/(1 +3x)  which tends to 3/(1  + 0)    = 3

Second one, lim  xcot(x)    = lim x { cos(x)/sin(x)}   = lim  { x cos (x)}  /sin (x)

divide numerator and denomimator by x   , lim  = cos(x) /  { sin(x) /x }

Now,{sin(x)}/x  tends to 1 as x tends to zero    so lim  = cos(x) /1  as x tends to zero,which is   1   since cos zero =1