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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits4 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{n}\rightarrow\infty}{\Big(\frac{1}{\text{n}^2}+\frac{2}{\text{n}^2}+\frac{3}{\text{n}^2}+\ \cdots+\frac{\text{n}-1}{\text{n}^2}}{}\Big)$
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Answer

$\lim\limits_{\text{n}\rightarrow\infty}{\Big(\frac{1}{\text{n}^2}+\frac{2}{\text{n}^2}+\frac{3}{\text{n}^2}+\ \cdots+\frac{\text{n}-1}{\text{n}^2}}{}\Big)$$=\lim\limits_{\text{n}\rightarrow\infty}\Big(\frac{1+2+3+\ \cdots+(\text{n}-1)}{{n}^2}\Big)$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{\big(\text{n}+1\big)(\text{n})}{2\times\text{n}^2}$ $\Big[1+2+3+\ \cdots+({\text{n}}-1)=\frac{({\text{n}}-1)({\text{n}})}{2}\Big]$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{\text{n}^2-\text{n}}{2\text{n}^2}$ $\Big[\frac{\infty}{\infty}\text{ from}\Big]$
$=\lim\limits_{\text{n}\rightarrow{\infty}}\frac{1-\frac{1}{\text{n}}}{2}$
$=\frac{1-0}{2}=\frac12$
$=\frac{1}{2}$

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