\begin{matrix} lim\\ x \to 0 \end{matrix}\frac{x+2\sin x}{\sqrt{x^{2}+2\sin x+1}-\sqrt{\sin ^{2}x-x+1}} is:

 

  • Option 1)

    6

  • Option 2)

    2

  • Option 3)

    3

  • Option 4)

    1

 

Answers (1)

\begin{matrix} lim\\ x \to 0 \end{matrix}\frac{x+2\sin x}{\sqrt{x^{2}+2\sin x+1}-\sqrt{\sin ^{2}x-x+1}}

=>\begin{matrix} lim\\ x \to 0 \end{matrix}\frac{(x+2\sin x)(\sqrt{x^{2}+2sinx+1}+\sqrt{sin^{2}x-x+1})}{(\sqrt{x^{2}+2\sin x+1})^{2}-(\sqrt{\sin ^{2}x-x+1})^{2}}

=>\begin{matrix} lim\\ x \to 0 \end{matrix}\frac{(x+2\sin x)(\sqrt{x^{2}+2sinx+1}+\sqrt{1-cos^{2}x-x+1})}{({x^{2}+2\sin x+1})-({\sin ^{2}x-x+1})}

=>\begin{matrix} lim\\ x \to 0 \end{matrix}\frac{(x+2\sin x)(\sqrt{x^{2}+2sinx+1}+\sqrt{2-cos^{2}x-x})}{x^{2}+2\sin x-\sin ^{2}x+x}

=>\begin{matrix} lim\\ x \to 0 \end{matrix}\frac{(x+2\sin x)(2)}{x^{2}+2\sin x-\sin ^{2}x+x}

\frac{0}{0}\;\;form\;\;use\;\;L'Hopital\;rule

\\\Rightarrow \lim_{x\rightarrow 0}\frac{(1+2\cos x)\times2}{2x+2\cos x-2\sin x\cos x+1}

=>\frac{(1+2)(2)}{0+1+2-0}

=>2


Option 1)

6

Option 2)

2

Option 3)

3

Option 4)

1

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