No. for (a) you have lim 2 when x --> 1- since F(x) = 2 for -1 x < 1.
(b) is right. (c) is "doesn't exist", since lim F(x) when x --> -1- doesn't equal lim F(x) = -1+: the former is 1/-1 = -1, whereas the latter is 2.
Also (d) is incorrect: the limit exists since both one-sided limits equal 2. That none of these equals F(1) only means F(x) isn't continuous at x = 1, but still the limit exists.
(e) is correct but for the reasons above, not for what you thought.