Well, your images are sideways and a bit hard to read, but they appear to all be the same. Was that intentional? In any case, here's what I think the problem is, for the benefit of future readers:
\(\displaystyle \displaystyle \lim_{x \to 0^{\pm}} \left( \arctan \left( e^{\frac{1}{x}} \right) \right)\)
Further, I'm guessing that the \(\displaystyle \pm\) is meant to indicate this is a two-sided limit, approaching from both above and below. But if that's the case, this problem is literally unsolvable, because the left-hand limit is different from the right-hand limit. Graph the function, using a graphing calculator (I prefer
Desmoshttps://www.desmos.com/calculator) and you'll see that as x approaches 0 from below, the function appears to be closing in on 0. But as x approaches 0 from above, the function appears to be closing in on some value in between 1 and 2. Clearly these are not the same value, so the two sided limit does not exist.
Please reply with any necessary corrections that make the exercise solvable, conferring with your instructor if necessary. Thank you.
