Apply the theorem: /Let (XN) be a sequence of positive real Numbers such that Li 2:= lim (XN+1/XN) exists. If L<1, then (XN) converges and lim (XN) = 0. to the following sequences, where a,b satisfy O

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Q: Given f(x)= ((n+1)(n+2)(n+3)....(n+2023))^(1/2023) then find lim n approaches to infinity f(x)(means…

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Q: 3. Let (Sn) be a sequence of nonnegative real numbers, and suppose lim s, = 0. Prove that lim S, =…

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Q: f(x) = x(x-1) for rational numbers          = 0 for irrational numbers  Prove f is continuous at x…

A: Given function is continuous at x= 0 and x=1. See the attachment.

Q: Prove the following. (b) lim( Vn² +1 – n ") = 0 - (c) lim Vn? n) = } 1 2 ||

A: Prove the following limits

Q: 6. (a) A sequence {anneN is said to be bounded if there exists M > 0 such that an M for any n E N…

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Q: If f is continuous and {an} is a sequence such that the limit lim an = L n➔ oo exists, then lim f…

A: Given: f is continuous at L.  an is an sequence such that limn→∞ an=L exists Now we have to prove…

Q: correspondig limit x' +x -8x-12 a) lím x* +4x' +5x + 4x +4

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Q: 3. Prove that for any bounded sequences (n) and yn we have lim sup(n + yn) ≤ lim sup(x) + lim…

A: We have to prove that limsupxn+yn≤limsupxn+limsupyn. Let xn and yn are sequence of R. Let limsupxn=x…

Q: lim [ln(1 + x²) − ln(1 + x)] - X

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