3 Use the 2-3 definition of lim f(x)=1 to prove that lim x+8 = 14.
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
definition of limit to prove that lim ,-e3. 3, (a) Use the - (b) Suppose lim g(z) 0 and if(x)| |g(z)| for all z E R. Use the ε-δ definition of limit to prove that lim f(x)=0 definition of limit to prove that lim ,-e3. 3, (a) Use the - (b) Suppose lim g(z) 0 and if(x)| |g(z)| for all z E R. Use the ε-δ definition of limit to prove that lim f(x)=0
2n 3. Prove that lim n+on+ 1 2.
(3) Use the definition of convergence to prove each of the following (a) 1 is not the limit of the sequence sn (-1)" (b) lim = 1/2 2n (c) Suppose that lim an = a. Prove that lim 3 . an За.
Use the definition to prove lim x +4 - X -1 2x + 1
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
m2 2. Prove that lim -+0n3 + 1 -=0. 3 5 100 3n2 + 2n - 1 3. Prove that lim = 5n2 +8 cos(n) 4. Prove that lim = 0. n-700 m2 + 17 5. Prove that lim (Vn+1 - Vn) = 0 Hint: Multiply Vn+1-vñ by 1 in a useful way. In particular, multiply Vn+1-17 by Vn+1+vn
question 3 and 4 (3) Prove that nen! T($) = lim n70s(s+1)...(s+n) (4) (a) Prove Wallis's formula T_2.2 4.4 2m. 2m 2" 1.3 3.5 (2m – 1)(2m + 1) *** (b) Use it to prove the identity t(s)[(s +1/2) = V721–25T(28)
Question 14 Evaluate: lim,-1+ [(222 - 1) = -1)] 2.0 se - 1 DA