Use squeeze theorem
an=n^2sin(n)/3n^4+5
Any doubt in steps then comment below.. I will explain you.
Here , I confuse that 5 is in denominator or not....so I. Take both case...you pick that one which is given in question..
Use the Main Limit Theorem (see Theorem 2.3.6) to prove that 4n2-3n-7 4 3n2 2n+5 3
(complete the proof. Hint: Use the Squeeze Theorem to show that lima = L.) 3- For all ne N, let an = Let S = {a, neN). 3-1) Use the fact that lim 0 and the result of Exercise 1 to show that OES'. 3-2) Use the result of Exercise 2 to show that S - {0}. 4- Prove that
2) Prove convergence of In} using the Squeeze lemma: n2-3n+7 a) In = 73 - 4n - 1
(1) Use the Squeeze Theorem to show that limx-ox* cos(207x) = 0. Give all your reasons. (2) Use ONLY THE DEFINITION (Either f'(a) = lim - 12)={@ or f'(a) = lima_vo f(a+h)-f(@)) to find the derivative of f(x) = 2x +1 at x = 3. (3) Findf, if f'(x) = 2+1 +20 +€* – sin 2 and f(0) = In(5). (4) Differentiate y = x*
Use the definition of 0 to show that 5n^5 +4n^4 + 3n^3 + 2n^2 + n 0(n^5).Use the definition of 0 to show that 2n^2 - n+ 3 0(n^2).Let f,g,h : N 1R*. Use the definition of big-Oh to prove that if/(n) 6 0(g{n)) and g(n) 0(h{n)) then/(n) 0(/i(n)). You should use different letters for the constants (i.e. don't use c to denote the constant for each big-Oh).
5) Test the series for convergence or divergence. n a) In 3n +1 n= b) cos(3n) 1+ (1.2)" n=1
Use a recursive tree method for recurrence function T(n)= 2T(n/5)+3n. then use substitution method to verify your answer
Using the complex-n-th roots theorem:
5. (a) Use Theorem 10.5.1: Complex n-th Roots Theorem (CNRT) to com- pute all the 4-th roots of -1/4. (b) Factor the polynomial 4x4 + 1 in C[x]. (c) Factor the polynomial 4x4 +1 in R[x]. (d) Use Rational Roots Theorem to prove that the polynomial 4x4 + 1 has no rational roots. Deduce the factorization of 4x4 + 1 in Q[x].
Prove the following: 1+4+7+...+(3n – 2) n(3n-1) 2
(Proof of the Squeeze Theorem for Functional Limits). Let f.g, h: A R be three functions satisfying f(x) < 9(2) < h(r) for all re A, and suppose c is a limit point of A and lim; cf(x) = L and lim -ch() = L. Prove that lim.+c9(x) = L as well.