Inverse Champernowne Substrings
MATL, 37 bytes
Thanks to @AndrasDeak for his help with Octsympy's vpa function!
'1/'1e4:&quot;@V]N$h1e4H$Y$t32&gt;)I0h)jXf1)q

EDIT (June 11, 2016): due to changes in the language, replace 0 by J in the code. The link below includes that modification
Try it online!
It takes a while in the online compiler (less than 1 minute).
'1/'        % literal string
1e4:        % array [1,2,...,1e4]
&quot;           % for each number in that array
  @V        % push number and convert to string
]           % end loop
N$h         % concatenate all strings
1e4H$Y$     % compute 1/123456789101112... as a string with 1e4 significant digits
t32&gt;)       % remove unwanted spaces and newlines in the output string
I0h)        % remove the first two characters ('8.')
j           % input string
Xf          % find indices of occurrences of input string within computed string
1)          % take first index
q           % subtract 1