Word Count : 191

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We are given the limit: \[ \lim_{n\to\infty} \left[\frac{n+1}{n^2+1^2}+\frac{n+2}{n^2+2^2}+\frac{n+3}{n^2+3^2}+\cdots+\frac{n+n}{n^2+n^2}\right] \] ### Step 1: Convert to a General Form The given sum can be written as: \[ S_n = \sum_{k=1}^{n} \frac{n+k}{n^2+k^2} \] Rewriting the fraction: \[ \frac{n+k}{n^2+k^2} = \frac{n+k}{n^2(1 + \frac{k^2}{n^2})} \] For large \( n \), approximate \( ________ ________ ______ _______ ____ _________ _______ ______ _________ _______ _____.
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