Could someone please check my work

Show that the equation 3 = x has at least one real solution. Since 3* = a then 3* - x2 = 0 Let f(a) = 30 - 202 =0 can say This by not sure I Theorem 4.2. 1 Suppose an - C . Then by Theorem 5.1.13, Intog boxl s and lim f(Xn) = lim 3t - (xn)? = 3 [limon) - [ lim En] by Theorem 4.2.1 = 30 - c2 = f(c) Thus, by Theorem 5.2.2, f is continuous at c BUT Laws 2.2.1 THEOREM Suppose that (Sn) and (tn) are convergent sequences with lim s,, = s and f ( - 1) = 3-1 - (-1)2 = - Co/ N lim th = t. Then (a) lim (Sn + tn) = s+ t. f(1) = 31 - (1)2 = 2 (b) lim (ks,) = ks and lim (k + s,) = k + s, for any ke R. (c) lim (Sntn) = st. Thus, f( - 1)