The most simple workaround to this problem is to create a buffer around your line, then you convert the buffer into a line ("Feature to line" using both the buffers and the original lines) and finally you run the "extend" tool (Info/advanced licence needed for feature to line, and Editor/standard licence needed for extend, but there are free solutions for the feature to line tool).
EDIT (Thanks Hornbydd for the remarks): you can directly create the line buffer in an edit session (editor toolbar > Editor > Buffer). This tool will work with selected feature only. The extend tool is also available as an editing tool (editor toolbar > Editor > more editing tools > advanced editing)