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Limits and Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits and Derivatives5 Marks
Question
Evaluate:
$\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}+3\text{x}}{2\text{x}+\tan3\text{x}}$
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Answer

Given that $\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}+3\text{x}}{2\text{x}+\tan3\text{x}}$
$=\lim\limits_{\text{x} \rightarrow 0}\frac{\big(\frac{\sin2\text{x}+3\text{x}}{2\text{x}}\big)\times2\text{x}}{\big(\frac{2\text{x}+\tan3\text{x}}{3\text{x}}\big)\times3\text{x}}$
$=\lim\limits_{\text{x} \rightarrow 0}\frac{\Big(\frac{\sin2\text{x}}{2\text{x}}+\frac{3\text{x}}{2\text{x}}\Big)\times2\text{x}}{\Big(\frac{2\text{x}\tan3\text{x}}{3\text{x}}\Big)\times3\text{x}}$
$=\lim\limits_{\text{x} \rightarrow 0}\frac{\Big(\frac{\sin2\text{x}}{2\text{x}}+\frac{3\text{x}}{2\text{x}}\Big)\times2\text{x}}{\Big(\frac{2\text{x}}{3\text{x}}+\frac{\tan3\text{x}}{3\text{x}}\Big)\times3\text{x}}$
$=\frac{\Big(\lim\limits_{2\text{x} \rightarrow 0}\frac{\sin2\text{x}}{2\text{x}}+\frac{3}{2}\Big)}{\Big(\frac{2}{3}+\lim\limits_{3\text{x} \rightarrow 0}\frac{\tan3\text{x}}{3\text{x}}\Big)}\times\frac{2}{3}$
$=\Big(\frac{1+\frac{3}{2}}{\frac{2}{3}+1}\Big)\times\frac{2}{3}$
$=\frac{\frac{5}{2}}{\frac{5}{3}}\times\frac{2}{3}=\frac{3}{2}\times\frac{2}{3}=1$
Hence, the required answer is 1.

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