Question

lim X tends to 0 Xsin2X/tan²3X​

Answer 1

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•Question:

$\red\bigstar \large\rm{ lim_{x \to 0} \: \: \dfrac{xsin2x}{tan^2 3x}}$

•To find :

$\red\bigstar\large\rm{lim_{ x \to 0} \: =?}$

•Given:

$\red\bigstar\large\rm{ lim_{ x \to 0} \: \: \dfrac{xsin2x}{tan^2 3x}}$

•Solution :

$\\ \implies\displaystyle\large\sf { lim_{ x \to 0} \: \: \dfrac{ x sin2x}{tan^2 3x}}$

$\\ \implies\displaystyle\large\sf{ lim_{ x \to 0 } \: \: \dfrac{ \dfrac{ x sin2x \: \times 2x}{2x}}{\dfrac{tan^2 3x \: \times 3x }{3x}}}$

•Properties :

$\\ \pink\star\large\rm{lim_{ x \to 0} \: \: \dfrac{tanx}{x}=1}$

$\\ \pink\star\large\rm{lim_{ x \to 0} \: \: \dfrac{sinx}{x}=1}$

$\\ \pink\star\large\rm{ lim_{x \to 0} \: \: \dfrac{cosx}{x} =1}$

•Put the properties In given equation :

$\\ \implies\displaystyle\large\sf{ lim_{ x \to 0}\: \: \dfrac{\dfrac{x sin2x \: \times 2x}{2x}}{\dfrac{tan^2 3x \: \times 3x}{3x}}}$

$\\ \implies\displaystyle\large\sf{ lim_{ x \to 0} \: \: \dfrac{x \times 2x }{3x}}$

$\\ \implies\displaystyle\large\sf{ lim_{ x \to 0} \: \: \dfrac{2x^2}{3x}}$

$\\ \implies\displaystyle\large\sf{ lim_{ x \to 0} \: \: \dfrac{2 \cancel{x^2}}{3 \cancel{x}}}$

$\\ \implies\displaystyle\large\sf{ lim_{x \to 0} \: \: \dfrac{ 2x}{3}}$

•Put the value of x=0 , in given equation .

$\\ \implies\displaystyle\large\sf{ lim_{x \to 0}\: \: \dfrac{ 2 \times 0}{3}}$

$\\ \implies\displaystyle\large\sf { lim_{ x \to 0} \: \: =0}$

$\pink\bigstar\large\boxed{\underline{\underline{\bf{ The \: correct \: answer \: will \: be \: 0}}}}$

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Methods to solving this question :

1. First check the function is $\dfrac{0}{0}$ form or not
2. If their is $\dfrac{0}{0}$form , use L-hospital rule (In L-hospital Rule , $\dfrac{0}{0}$ form removed.
3. Also this question is solve by using first principle of differentiation .