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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits2 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0^+}\Big\{1+\tan^2\sqrt{\text{x}}\Big\}^{\frac{1}{2}\text{x}}$
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Answer

$\lim\limits_{\text{x}\rightarrow0^+}\Big\{1+\tan^2\sqrt{\text{x}}\Big\}^{\frac{1}{2}\text{x}}$ $=\ \text{e}^{{\lim\limits_{\text{x}\rightarrow0^+}}\Big\{\frac{\tan^2\sqrt{\text{x}}}{2\text{x}}\Big\}}$ $=\ \text{e}^{{\lim\limits_{\text{x}\rightarrow0^+}}\Big\{\frac{\tan^2\sqrt{\text{x}}}{2\text{x}}\Big\}}$ $=\ \text{e}^{{\lim\limits_{\text{x}\rightarrow0^+}}\Big\{\frac{\sin^2\sqrt{\text{x}}}{2\text{x}\cos^2\sqrt{\text{x}}}\Big\}}$ $=\ \text{e}^{\frac{1}{2}\lim\limits_{\text{x}\rightarrow0^+}\bigg\{\Big(\frac{\sin\sqrt{\text{x}}}{\sqrt{\text{x}}}\Big)^2\bigg\}\lim\limits_{\text{x}\rightarrow0^+}\Big\{\frac{1}{\cos^2\sqrt{\text{x}}}\Big\}}$ $=\ \text{e}^\frac{1}{2}$ $=\ \sqrt{\text{e}}$

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