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Hello, can someone who know this help me?

 

\(\lim _{x\to \left(\pi \div 2\right)}\left(sin^2x\div \:cos^2x\right)\cdot \left(sinx-1\right)\)

 Jun 27, 2019
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Hello, can someone who know this help me?
\(\mathbf{ \lim \limits_{x\to \dfrac{\pi}{2}}\left(\dfrac{\sin^2(x)}{\cos^2(x)}\cdot \Big(\sin(x)-1\Big)\right) }\)

 

\(\begin{array}{|rcll|} \hline && \mathbf{ \lim \limits_{x\to \dfrac{\pi}{2}}\left(\dfrac{\sin^2(x)}{\cos^2(x)}\cdot \Big(\sin(x)-1\Big)\right) } \quad & | \quad \cos^2(x)=1-\sin^2(x) \\\\ &=& \lim \limits_{x\to \dfrac{\pi}{2}}\left(\dfrac{\sin^2(x)}{1-\sin^2(x)}\cdot \Big(\sin(x)-1\Big)\right) \\\\ &=& \lim \limits_{x\to \dfrac{\pi}{2}}\left(\dfrac{\sin^2(x)}{\Big(1-\sin(x)\Big)\Big(1+\sin(x)\Big)}\cdot \Big(\sin(x)-1\Big)\right) \\\\ &=& \lim \limits_{x\to \dfrac{\pi}{2}}\left(\dfrac{-\sin^2(x)}{\Big(1-\sin(x)\Big)\Big(1+\sin(x)\Big)}\cdot \Big(1-\sin(x)\Big)\right) \\\\ &=& \lim \limits_{x\to \dfrac{\pi}{2}}\left(\dfrac{-\sin^2(x)}{\Big(1+\sin(x)\Big)}\right) \\\\ &=& \dfrac{-\sin^2(\dfrac{\pi}{2})}{\Big(1+\sin\left(\dfrac{\pi}{2}\right)\Big)} \quad & | \quad \sin\left(\dfrac{\pi}{2}\right) = 1 \\\\ &=& \dfrac{-1}{ 1+1 } \\\\ &=& \mathbf{ -\dfrac{1}{ 2 } } \\ \hline \end{array}\)

 

laugh

 Jun 27, 2019

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