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Limits — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits5 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\infty}\Big\{\frac{\text{x}^2+2\text{x}+3}{2\text{x}^2+\text{x}+5}\Big\}^{\frac{3\text{x}-2}{3\text{x}+2}}$
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Answer

$\lim\limits_{\text{x}\rightarrow\infty}\Big\{\frac{\text{x}^2+2\text{x}+3}{2\text{x}^2+\text{x}+5}\Big\}^{\frac{3\text{x}-2}{3\text{x}+2}}$
$=\text{e}^{\lim\limits_{\text{x}\rightarrow\infty}\Big\{\Big({\frac{3\text{x}-2}{3\text{x}+2}}\Big)\text{In}\Big(\frac{\text{x}^2+2\text{x}+3}{2\text{x}^2+\text{x}+5}\Big)\Big\}}$
$=\text{e}^{\lim\limits_{\text{x}\rightarrow\infty}\Bigg\{\Bigg(\frac{3-\frac{2}{\text{x}}}{3+\frac{2}{\text{x}}}\Bigg)\Big(\text{In}\Big(\frac{\text{x}^2+2\text{x}+3}{2\text{x}^2+\text{x}+5}\Big)\Big)\Bigg\}}$
$=\text{e}^{\lim\limits_{\text{x}\rightarrow\infty}\Bigg\{\Bigg(\frac{3-\frac{2}{\text{x}}}{3+\frac{2}{\text{x}}}\Bigg)\begin{pmatrix}\text{In}\begin{pmatrix}\frac{1+\frac{2}{\text{x}}+\frac{3}{\text{x}^2}}{2+\frac{1}{\text{x}}+\frac{5}{\text{x}^2}}\end{pmatrix}\end{pmatrix}\Bigg\}}$
$=\text{e}^{1.\text{In}\big(\frac{1}{2}\big)}$
$=\frac{1}{2}$

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