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Limits — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSLimits5 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{n}\rightarrow\infty}\frac{{(\text{n}+2)!}+{(\text{n}+1)!}}{{(\text{n}+2)!}+{(\text{n}+1)!}}$
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Answer

$\lim\limits_{\text{n}\rightarrow\infty}\frac{{(\text{n}+2)!}+{(\text{n}+1)!}}{{(\text{n}+2)!}+{(\text{n}+1)!}}$
We know that (n + 2) = (n + 2)(n + 1)!
$\Rightarrow\lim\limits_{\text{n}\rightarrow\infty}\frac{(\text{n}+2)(\text{n}+1)!+(\text{n}+1)!}{(\text{n}+2)(\text{n}+1)!-(\text{n}+1)!}$ 
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{(\text{n}+1)!\big[(\text{n}+2)+1\big]}{(\text{n}+1)\big[(\text{n}+2)-1\big]}$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{\text{n}+3}{\text{n}+1}$ $\Big[\frac\infty\infty\text{ from}\Big]$
$=\lim\limits_{\text{n}\rightarrow{\infty}}\frac{1+\frac{3}{\text{n}}}{1+\frac{1}{\text{n}}}$
$=\frac{1+0}{1+0}$
$=1$

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