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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits2 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{3\sin\text{x}-4\sin^3\text{x}}{\text{x}}$
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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{3\sin\text{x}-4\sin^3\text{x}}{\text{x}}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{\sin3\text{x}}{{\text{x}}}$ $\big[\because\sin3\text{x}=3\sin\text{x}-4\sin^3\text{x}\big]$ $=\lim\limits_{\text{x}\rightarrow0}\frac{\sin3\text{x}}{{3\text{x}}}\times3$ $=3\times\lim\limits_{\text{x}\rightarrow0}\frac{\sin3\text{x}}{{3\text{x}}}$ $=3\times\lim\limits_{3\text{x}\rightarrow0}\frac{\sin3\text{x}}{{3\text{x}}}$ $\big[\because\text{x}\rightarrow0,3\text{x}\rightarrow0\big]$ $=3\times1$ $\Big[\because\ \lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1\Big]$ $=3$

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