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Question
`lim_{n→∞}[1/(n^2 + 1) + 2/(n^2 + 1) + 3/(n^2 + 1) + .... + n/(n^2 + 1)]` = ______
Options
1
2
`1/2`
0
MCQ
Fill in the Blanks
Solution
`lim_{n→∞}[1/(n^2 + 1) + 2/(n^2 + 1) + 3/(n^2 + 1) + .... + n/(n^2 + 1)]` = `underline(1/2)`
Explanation:
Given limit = `lim_{n→∞} (1 + 2 + 3 + ... + n)/(1 + n^2)`
= `lim_{n→∞} (sumn)/(1 + n^2)`
= `lim_{n→∞} 1/2 (n(n + 1))/(1 + n^2)`
= `lim_{n→∞} 1/2 ((1 + 1/n))/((1/n^2 + 1))`
= `1/2. 1 = 1/2`
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Substitution Method
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