JEE MainMathematicsModerateLimits

$\lim_{x \rightarrow 0}\mspace{2mu}\frac{ln\left( tan\left( \frac{\pi}{4} + ax \right) \right)}{sinbx},b \neq 0$
is equal to

Show answer and solution

\lim_{x \rightarrow 0}\mspace{2mu}\frac{\mathcal{l}n\left( tan\left( \frac{\pi}{4} + ax \right) \right)}{sinbx},b \neq 0

\lim_{x \rightarrow 0}\mspace{2mu}\frac{\mathcal{l}n\left( 1 + \frac{2tanax}{1 - tanax} \right) \cdot \left( \frac{2tanax}{1 - tanax} \right)(bx)}{\left( \frac{2tanax}{1 - tanax} \right)}sinh(bx)

\lim_{x \rightarrow 0}\mspace{2mu}\frac{2tanax}{bx} = \frac{2a}{b}

Source: Grass-G / M-19 Mathematics Paper 28 Nov Fourth.docx

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