problem 3 SEL 3. Prove that the mapping w 4. Prove that w z3 3z 1 is one-to-one for |z <1 Z n S {z| |z < 1} is continuous at z 1 +z6 5. Find the lim
IDY in < oo and lim - Yn < 0o. Prove that lim,+ 1. Let In > 0. Yn > 0 such that lim,- Yn) < lim,-- In lim,+ Yn: i tn < oo and lim yn < . Prove that lim. In 1. Let In 20, yn 0 such that lim Yn) < limn+In lim + Yr
2. Prove that lim (-1)"+1 0. 72-00 n 2n 3. Prove that lim noon + 1 2. 80 4. Prove that lim n-+v5n 0. -7 9 - in 5. Prove that lim n0 8 + 13n 13
1. Determine if the following limits exist. In each case prove and explain your argument. (a) lim x+y + xy sin x siny x²y lim *+(0,0) x4 + y2 x(0,0) xy
(c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a < x < b.) (c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a
1. Determine if the following limits exist. In each case prove and explain your argument. (c) lim x +y + y sin x siny *(0,0) XY lim x-(0,0) x4 + y2 lim x+(0,0) x2y2 + (x + y2)2
3. (10 marks) Find the limit and prove it using the definition. 4x2 + 13 lim x+ x2 + x + 1 4. (10 marks) Find the limit and prove it using the definition. 4x3 + 13 lim *40x2 + x + 1
Let f be defined on an open interval I containing a point a (1) Prove that if f is differentiable on I and f"(a) exists, then lim h-+0 (a 2 h2 (2) Prove that if f is continuous at a and there exist constants α and β such that the limit L := lim h2 exists, then f(a)-α and f'(a)-β. Does f"(a) exist and equal to 2L? Let f be defined on an open interval I containing a point a...
5.2.6 Prove Theorem 5.12 using a sequential argument. Theorem 5.12 (Boundedness Away from Zero) If the limit lim f(x) +20 exists and is not zero, then there is an interval (xo – C, X0 + c) and a positive number m such that \f(x) > m > 0 for every value of x + xo in that interval and that belongs to the domain of f.
3) Find the limit and prove it using definition lim 4x² + 13 x 70 x² + xt I