Could someone please check my work and make sure both cases are correct

Please state all definitions and theorems that you will need: Theorem 5.1.13 Let f: D -+ R and g: D -+ R and let c be an accumulation pint of D . If lim f(x) = L , lim g(x) = M , and k E R , then lim (f + 9)(x) = L + M, lim (fg)(x) = LM . and lim (kf)(z) = KL . Furthermore, if g(a) * 0 for all x E D and M * 0 , then lim ( : ) (2 ) = . Theorem 4.2.1 Suppose that (Sn) and (tn) are convergent sequences with lim s,, = s and lim to = t . Then (a) lim (sn + tn) = s+ t (b) lim (ksn) = ks and lim (k + 8n) = k + s, for any k E IR (c) lim (sn . tn) = st (d) lim = > , provided that tn # 0 for all n and t # 0 Theorem 5.2.2 Let f : D - R and let c E D . Then the following three conditions are equivalent. (a) f is continuous at c (b) If (In) is any sequence in D such that (In) converges to c, then lim f(an) = f(c) (c) For every neighborhood V of f(c) there exists a neighborhood U of c such that f(UnD) CV . Furthermore, if c is an accumulation point of D , then the above are all equivalent to (d) f has a limit at c and lim f(x) = f(c) . Theorem 5.3.6 (Intermediate Value Theorem) Suppose that f: [a, b] - R is continuous. Then f has the intermediate value property on [a, b] . That is, if k is any value between f(a) and f(b) [i.e., f(a) a , then if f(a) = b, thenf(a) > a but if f ( b ) = a , then f ( b ) 0 and f(6) - b 0 and F (b ) = f(6 ) - 6