Question

(b) Evaluate the following limits or explain why they do not exist: Hint: you do not have to give e, 6 type arguments; use the properties given in the notes. (i) (1P.) 3z2i lim 2-1 z 4z2 + z (ii) (2P.) Re(z) lim Im(2)

Answer 1

o lim $(32+21) ZI 271 751 21 211 (23+4z2+z) ) : Let F (Z) = (32+21) f g (2) = (23+422 +2). Now, lim F(z)Lim (32+21) 4 lim G(z): lim (23+4 z?+2) = 3x1 +21) = (1 3+4 (02+0 =(3+21) 0 6 - We know that We know that tim [F (2) 2 sim [F(z)]. 2k [9 (2) J um racas) s lim { [32721') } = 372 lim (32+21)|tim (22+422+z) 271 L (23+422+2) J. 1 201 3+2 i) { by eq 04@? 27K TOK 271 6 tim (Re(z) L Ang (10 720 Im (2)) .: Z = (x try) also as 730 ,9 This question is li Re (Z) = x, I'm (2)= y. 0 and yoz o. x 90 y > 0

Now take limit along the line y=mx. lim I? lim 20 Thus since limit depends on variable im': Hence Limit does not exist Ary.