Question

Lt. sec x-2
x-π/3. x-π/3 Is equal to​

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{\displaystyle\lim_{x\to\;\frac{\pi}{3}}\;\dfrac{sec\,x-2}{x-\dfrac{\pi}{3}}}$

$\underline{\textbf{To find:}}$

$\textsf{The value of}$

$\mathsf{\displaystyle\lim_{x\to\;\frac{\pi}{3}}\;\dfrac{sec\,x-2}{x-\dfrac{\pi}{3}}}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{\displaystyle\lim_{x\to\;\frac{\pi}{3}}\;\dfrac{sec\,x-2}{x-\dfrac{\pi}{3}}}$

$\mathsf{=\dfrac{sec\,\dfrac{\pi}{3}-2}{\dfrac{\pi}{3}-\dfrac{\pi}{3}}}$

$\mathsf{=\dfrac{0}{0}\;form}$

$\textsf{Apply L Hopital's rule,}$

$\mathsf{=\displaystyle\lim_{x\to\;\frac{\pi}{3}}\;\dfrac{sec\,x\;tan\,x}{1-0}}$

$\mathsf{=sec\dfrac{\pi}{3}\;tan\dfrac{\pi}{3}}$

$\mathsf{=2\,\sqrt{3}}$

$\implies\boxed{\mathsf{\displaystyle\lim_{x\to\;\frac{\pi}{3}}\;\dfrac{sec\,x-2}{x-\dfrac{\pi}{3}}=2\,\sqrt{3}}}$

$\underline{\textbf{Find more:}}$

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