fyi,

Mathematica 8.0 for Mac OS X x86 (64-bit)
Copyright 1988-2011 Wolfram Research, Inc.

In[1]:= Limit[Sum[Sin[Pi*k/n]/(n + 1/k), {k, 1, n}], n -> Infinity] 

        2
Out[1]= --
        Pi

In[2]:= $Version

Out[2]= 8.0 for Mac OS X x86 (64-bit) (February 23, 2011)


Mathematica 10.0 for Mac OS X x86 (64-bit)
Copyright 1988-2014 Wolfram Research, Inc.

In[1]:= Limit[Sum[Sin[Pi*k/n]/(n + 1/k), {k, 1, n}], n -> Infinity]                     

               -I     (I Pi)/n       ((2 I) Pi)/n       (-I Pi)/n -1 + n
Out[1]= Limit[(-- (-(E         n) - E             n + (E         )       n +    ... 
                2 

In[2]:= $Version                                                                        

Out[2]= 10.0 for Mac OS X x86 (64-bit) (September 10, 2014)

The 10.0 result is not exactly wrong. This sort of back step sometimes happens when a similar input gave an incorrect result and the fix disabled related functionality.

I can ask developers about this specific example.