Question

find the limit by applying l hospital rule.​I want full solution

Answer 1

Given Question

Find limit by applying L - Hospital Rule

$\rm :\longmapsto\:\displaystyle\lim _{x \to 0}\rm \dfrac{x(1 - \sqrt{1 - {x}^{2}})}{ \sqrt{1 - {x}^{2}} \: ({sin}^{ - 1} ( {x}^{3}) )}$

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

Given expression is

$\rm :\longmapsto\:\displaystyle\lim _{x \to 0}\rm \dfrac{x(1 - \sqrt{1 - {x}^{2}})}{ \sqrt{1 - {x}^{2}} \: ({sin}^{ - 1} ( {x}^{3}) )}$

can be rewritten as

$\rm \:  =  \: \displaystyle\lim _{x \to 0}\rm \frac{1}{ \sqrt{1 - {x}^{2} } } \times \displaystyle\lim _{x \to 0}\rm \frac{x(1 - \sqrt{1 - {x}^{2}})}{ {sin}^{ - 1} {x}^{3} }$

$\rm \:  =  \: 1 \times \displaystyle\lim _{x \to 0}\rm \frac{x(1 - \sqrt{1 - {x}^{2}})}{ \dfrac{ {sin}^{ - 1} {x}^{3}}{ {x}^{3} } \times {x}^{3} }$

We know,

$\purple{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\lim _{x \to 0}\rm \frac{ {sin}^{ - 1}x}{x} = 1}}}$

So, using this result, we get

$\rm \:  =  \: \displaystyle\lim _{x \to 0}\rm \frac{x(1 - \sqrt{1 - {x}^{2}})}{ {x}^{3} }$

$\rm \:  =  \: \displaystyle\lim _{x \to 0}\rm \frac{1 - \sqrt{1 - {x}^{2}}}{ {x}^{2} }$

Now, on applying L - Hospital Rule, we get

$\rm \:  =  \: \displaystyle\lim _{x \to 0}\rm \frac{\dfrac{d}{dx}(1 - \sqrt{1 - {x}^{2}})}{\dfrac{d}{dx} {x}^{2} }$

We know,

$\purple{\rm :\longmapsto\:\boxed{\tt{ \dfrac{d}{dx} {x}^{n} \: = \: {nx}^{n - 1} \: }}}$

and

$\purple{\rm :\longmapsto\:\boxed{\tt{ \dfrac{d}{dx} \sqrt{x} \: = \: \frac{1}{2 \sqrt{x} } \: }}}$

So, using this result, we get

$\rm \:  =  \: \displaystyle\lim _{x \to 0}\rm \frac{0 - \dfrac{1}{2 \sqrt{1 - {x}^{2} }}\dfrac{d}{dx}(1 - {x}^{2}) }{2x}$

$\rm \:  =  \: \displaystyle\lim _{x \to 0}\rm \frac{ - \dfrac{1}{2 \sqrt{1 - {x}^{2} }}(0 - 2x) }{2x}$

$\rm \:  =  \: \displaystyle\lim _{x \to 0}\rm \frac{1}{2 \sqrt{1 - {x}^{2} } }$

$\rm \:  =  \: \dfrac{1}{2}$

Hence,

$\red{\rm\implies \:\:\boxed{\tt{ \displaystyle\lim _{x \to 0}\rm \dfrac{x(1 - \sqrt{1 - {x}^{2}})}{ \sqrt{1 - {x}^{2}} \: ({sin}^{ - 1} ( {x}^{3}) )} = \frac{1}{2}}}}$

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MORE TO KNOW

$\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {e}^{x} & \sf {e}^{x} \end{array}} \\ \end{gathered}$

Answer 2

Answer:

$\boxed{{\mathbb\pink{REFERR \:TO\: THE\:\: ATTACHMENT }}}$

Step-by-step explanation:

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