Joe gave the following argument: since lim x→0 0 = 0,
(a) and since 0 = −1 x + 1 x ,
(b) we know that lim x→0 ( −1 x + 1 x ) = 0.
(c) but then, since lim x→0 ( −1 x + 1 x ) = lim x→0 ( −1 x ) + lim x→0 ( 1 x ),
(d) we can say that lim x→0 ( −1 x ) + lim x→0 ( 1 x ) = 0, which means that lim x→0 ( −1 x ) = − lim x→0 ( 1 x ).
(e) no error
in which line, if any, has joe made an error?