Question

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Problem 1. Let {r,} be a sequence and L be a real number. Give the definition that lim, In L. Prove from the definition of the limit, that 2n2 + 1 lim nx 4n? - n + 1 %3D by completing the following steps. (a) Using the fact that 1 <n < n?, estimate from above the expression 2n? +1 4n2 – n+1 b) Given e > 0 find a threshold N, so that for all n > N: 2n2 + 1 < E. 4n2 –n +1

Answer 1

lim an² 4n²ntl Let an²+1 4n²nti show that that integer I which satisfy the limit of a is to find a N, for each t the inequality sequence positive o consider, if nz N. 2n²+1 Tan²-nti 4n²+2-4n²th-) 2 (4n²_nti) n + 2 (4n² n+1) n + ) 8n2_2n+2

NOW lanan2 2 2 ntlL n²+1 E 2 · ILE-1 VnN o Thus, for hinteger given t Ny we obtain a positive which holds Hablet that I Hence, this lim nco proves an²+1 4n²nt