As is, that's without a comma (,), sentence 1. could be ambiguous. 'using' this example' modifies the closest noun, unless, that is, it's separated from the nearest noun by a comma, and in that case it would modify the next closest noun. So, in 1., 'using this method' modifies the noun 'examples', whereas in 2. it modifies the (pro)noun 'they'.
1. They will show the examples using this method.
2. They will show the examples, using this method.
3. They will show, using this method, the examples.
Hope that helps.
Student or Learner