Question

lim x tends to 0 (xtanx)/(sin3x)

answer is 0 .

Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\:\displaystyle\lim_{x \to 0}\rm \frac{x \: tanx}{sin3x}$

If we substitute directly x = 0, we get

$\rm \: = \: \dfrac{0 \times tan0}{sin0}$

$\rm \: = \: \dfrac{0}{0}$

which is indeterminant form.

So,

$\rm :\longmapsto\:\displaystyle\lim_{x \to 0}\rm \frac{x \: tanx}{sin3x}$

$\rm \: = \: \displaystyle\lim_{x \to 0}\rm x \times \dfrac{tanx}{x} \times x \times \dfrac{3x}{sin3x} \times \dfrac{1}{3x}$

$\rm \: = \: \displaystyle\lim_{x \to 0}\rm x \times \displaystyle\lim_{x \to 0}\rm \frac{tanx}{x} \times \displaystyle\lim_{x \to 0}\rm \frac{3x}{sin3x} \times \frac{1}{3}$

We know,

$\purple{\rm :\longmapsto\:\boxed{\tt{ \: \: \: \displaystyle\lim_{x \to 0}\rm \frac{sinx}{x} = 1 \: \: \: }}}$

and

$\pink{\rm :\longmapsto\:\boxed{\tt{ \: \: \: \displaystyle\lim_{x \to 0}\rm \frac{tanx}{x} = 1 \: \: \: }}}$

So, using this, we get

$\rm \: = \: 0 \times 1 \times 1 \times \dfrac{1}{3}$

$\rm \: = \: 0$

Hence,

$\pink{\rm :\longmapsto\:\boxed{\tt{ \: \: \: \: \: \: \: \displaystyle\lim_{x \to 0}\rm \frac{x \: tanx}{sin3x} = 0 \: \: \: \: \: \: }}}$

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