R R 5. To compute 1 = lim 2 COS dr and J = lim 22+1 sinc dx simultaneously .22 +1 R R0 R R using Residue Theorem, let f(x) 22 +1 C COSC sinc (1) Show that if z = x + iy, then Rf(R2) = and Sf(R2) = x2 +1 x2 +1 (2) Find Res[f, i]. (3) Show that I = 0 and J (4) Prove I = 0) in the above problem without using Residue Theorem. IT
1. Compute the following limits algebraically, if possible: 4.2 +1-3 (a) lim 5.2 – 1 (b) lim 2 - 2 100 3.42 - 2.c 2 (c) lim csc(7x) sin(4x) (d) lim lim (1+3) 20 22 (e) lim +01 – COS X
1. Compute the following limits. 9x4 - 4y4 (a) lim lim (x,y)–(0,0) 3x2 + 2y2
2² +9 [10] 5. Compute lim +33 - 3
Part 1: Part 2: 1 2/2 Calculate lim +0 ( + ) Let f(x) be a differentiable function such that lim f(x) = 00. 200 In(f(x)) (a) Compute lim 10 f(x) e-2f(x) (b) Compute lim 2400 arctan( f(x)) –
Compute the limit by using known power series 2. lim 30 · (arctan (2) – sin (2)) 1 - cos (22)
2.13.2 Compute lim sups and lim infs for the following sequences (a) {(-1)"n} (b) {sin (nT/8)} (c) {nsin (nt/8)} (d) {[(n + 1) sin (n7/8)]/n} (e) {1+(-1)"} (f) {rn} consists of all rational numbers in the interval (0,1) arranged in some order.
please compute these two limits, or explain why they dont exist (2x - 2)y lim (x,y)+(1,0) (x - 1)2 + y2 xy2 lim (1,y)+(0,0) x2 + y2
8. Compute the limit arctan 2x lim +- X (a) 0 (b) 7/2 (c) –7/2 (d) e (e) 1
use taylor theorem 6. Using Taylor's theorem compute lim 1-1+2.2068