Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits4 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\frac{\pi}{3}}}\frac{\sqrt{3}-\tan\text{x}}{\pi-3\text{x}}$
✓
Answer
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{3}}}\frac{\sqrt{3}-\tan\text{x}}{\pi-3\text{x}}$ If $\text{x}\rightarrow\frac{\pi}{3},\frac{\pi}{3}-\text{x}\rightarrow0,\pi-3\text{x}\rightarrow0$ Let $\frac\pi3-\text{x}=\text{y}$ they y → 0 $=\lim\limits_{\text{y}\rightarrow{0}}\frac{\sqrt{3}-\tan\big(\frac{\pi}{3}-\text{y}\big)}{3\big(\frac{\pi}{3}-\text{x}\big)}$ $=\lim\limits_{\text{y}\rightarrow{0}}\begin{pmatrix}\frac{\bigg(\sqrt{3}-\frac{\tan\frac\pi3-\tan\text{y}}{1+\tan\frac\pi3.\tan\text{y}}\bigg)}{3\text{y}}\end{pmatrix}$ $=\lim\limits_{\text{y}\rightarrow{0}}\begin{pmatrix}\frac{\Big(\sqrt{3}-\frac{\sqrt{3}-\tan\text{y}}{1+\sqrt{3}\tan\text{y}}\Big)}{3\text{y}}\end{pmatrix}$ $=\lim\limits_{\text{y}\rightarrow{0}}\frac{\big(\sqrt{3}-\tan\text{y}-\sqrt{3}+\tan\text{y}\big)}{3\big(1+\sqrt{3}\tan\text{y}\big)\text{y}}$ $=\frac{4}{3}\times\lim\limits_{\text{y}\rightarrow0}\frac{\tan\text{y}}{\text{y}}\times\frac{1}{\lim\limits_{\text{y}\rightarrow0}\Big(1+\sqrt{3}\frac{\tan\text{y}}{\text{y}}\times\text{y}\Big)}$ $=\frac{4\times1}{3}\times\frac{1}{1+0}$ $=\frac{4}{3}$
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