csat-suneung 2025 Q25C

csat-suneung · South-Korea · csat__math 3 marks Sequences and series, recurrence and convergence Convergence proof and limit determination

For a sequence $\left\{ a_{n} \right\}$, $\lim_{n \rightarrow \infty} \frac{n a_{n}}{n^{2} + 3} = 1$. What is the value of $\lim_{n \rightarrow \infty} \left(\sqrt{a_{n}^{2} + n} - a_{n}\right)$? [3 points]
(1) $\frac{1}{3}$
(2) $\frac{1}{2}$
(3) $1$
(4) $2$
(5) $3$

For a sequence $\left\{ a_{n} \right\}$, $\lim_{n \rightarrow \infty} \frac{n a_{n}}{n^{2} + 3} = 1$. What is the value of $\lim_{n \rightarrow \infty} \left(\sqrt{a_{n}^{2} + n} - a_{n}\right)$? [3 points]\\
(1) $\frac{1}{3}$\\
(2) $\frac{1}{2}$\\
(3) $1$\\
(4) $2$\\
(5) $3$

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q23C Q23G Q24 Q24C Q25 Q25C Q26 Q26C Q27 Q27C Q27G Q28 Q28C Q28G Q29 Q29C Q29G Q30 Q30C Q30G