Question

lim x=o [(tan5x +6x) / tan3x+4x)]​

Answer 1

Limits

Formula:

$\displaystyle\quad\mathrm{\lim_{x\to 0}\frac{tan(mx)}{mx}=1}$

$\displaystyle\quad\mathrm{\lim_{x\to 0}(k)=k,\:k=constant}$

Solution:

$\displaystyle\quad\mathrm{\lim_{x\to 0}\frac{tan(5x)+6x}{tan(3x)+4x}}$

$\displaystyle\mathrm{=\lim_{x\to 0}\frac{\frac{tan(5x)+6x}{x}}{\frac{tan(3x)+4x}{x}}\quad[\because x\to 0\Rightarrow x\neq 0]}$

$\displaystyle\mathrm{=\lim_{x\to 0}\frac{\frac{tan(5x)}{x}+6}{\frac{tan(3x)}{x}+4}}$

$\mathrm{=\frac{\displaystyle \mathrm{5\lim_{x\to 0}\frac{tan(5x)}{5x}+6}}{\displaystyle \mathrm{3\lim_{x\to 0}\frac{tan(3x)}{3x}+4}}}$

$\displaystyle\mathrm{=\frac{5+6}{3+4}\quad[\because\lim_{x\to 0}\frac{tan(mx)}{mx}=1]}$

$\displaystyle\mathrm{=\frac{11}{7}}$

$\displaystyle\Rightarrow \boxed{\mathrm{\lim_{x\to 0}\frac{tan(5x)+6x}{tan(3x)+4x}=\frac{11}{7}}}$

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