I’m currently taking complex analysis and I have to take the limit for the following problem:

(see attachment)

but when I take the limit as -1 I have

Arg(-1) so if I compute it I have pi, but then there are some online resources out there that say that the limit of Arg(-1) is -pi

Wolfram alpha has this as pi as well

If Arg(z) and z = x+iy then

Arg(x+iy) and since -1 is a real number and there is no imaginary number then I have

Arg(-1)

But is the limit pi or -pi?