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- Prove or provide a counterexample to the following if r ∈ ( 0 , 1 ), then lim n→∞ r^n exists.Extend the result proved in Example 2.5.3 to the case |b| < 1;that is, show lim(bn) = 0 if and only if −1 < b < 1.In formally proving that limx→1 (x^2+x)=2, let ε>0 be arbitrary. Chooseδ =min(εm , 1). Determine the smallest value of m that would satisfy the proof.
- Show that if xn ≤ yn ≤ zn for all n ∈ N, and if lim xn = lim zn = l, then lim yn = l as well.Consider the following list of conjectures. Provide a shortproof for those that are true and a counterexample for any that are false. (a) If lim(an − bn) = 0, then lim an = lim bn.If (an) has limit −1 and (bn) tends to infinity, does it follow that only finitely many terms of (anbn) are positive?
- b) If (an) has limit−1 and (bn) tends to infinity, does it follow that only finitely many terms of (an bn) are positive?4. Let f RR and let cЄ R. Show that limf(x) = L if and only if lim f(x + c) = L. x-e x-0 PLEAEhelp me withthis and so the work!!In formally proving that lim x→1 (x2 + x) = 2, let ε > 0 be arbitrary. Choose δ = min (ε/m, 1).Determine the smallest value of m that would satisfy the proof.
- Compute the lim as x goes to 0+ (e^x - 1 - x)/ (xe^x - x)Use the limit Theorem 9.2-9.7 from ross eleentary analysis to prove that Justify allDetermine whether the following statement is true or false. If this statement is true, explain why. If this statement is false, provide a counterexample. If lim n→∞a_n = π, then a_n is divergent.