Soal yang Akan Dibahas
Nilai $ \displaystyle \lim_{x \to \frac{\pi}{4} }
\sin \left( \frac{\pi}{4} - x \right)\tan \left( x + \frac{\pi}{4} \right) $ adalah ....
A). $ 2 \, $ B). $ 1 \, $ C). $ 0 \, $ D). $ -1 \, $ E). $ -2 $
A). $ 2 \, $ B). $ 1 \, $ C). $ 0 \, $ D). $ -1 \, $ E). $ -2 $
$\spadesuit $ Konsep Dasar
*). \sifat Limit Trigonometri :
$ \displaystyle \lim_{x \to k} \frac{\sin f(x)}{\tan f(x)} = 1 $ ,
dengan syarat : $ f(k) = 0 $
*). Rumus Dasar Trigonometri :
$ \tan f(x) = \frac{1}{\cot f(x) } $
Sudut Komplemen : $ \cot A = \tan (\frac{\pi}{2} - A) $
*). \sifat Limit Trigonometri :
$ \displaystyle \lim_{x \to k} \frac{\sin f(x)}{\tan f(x)} = 1 $ ,
dengan syarat : $ f(k) = 0 $
*). Rumus Dasar Trigonometri :
$ \tan f(x) = \frac{1}{\cot f(x) } $
Sudut Komplemen : $ \cot A = \tan (\frac{\pi}{2} - A) $
$\clubsuit $ Pembahasan
*). Mengubah bentuk $ \tan \left( x + \frac{\pi}{4} \right) $ :
$\begin{align} \tan \left( x + \frac{\pi}{4} \right) & = \frac{1}{\cot \left( x + \frac{\pi}{4} \right) } \\ & = \frac{1}{\tan \left( \frac{\pi}{2} - ( x + \frac{\pi}{4} ) \right) } \\ & = \frac{1}{\tan \left( \frac{\pi}{4} - x \right) } \end{align} $
*). Menyelesaikan soal :
$\begin{align} & \displaystyle \lim_{x \to \frac{\pi}{4} } \sin \left( \frac{\pi}{4} - x \right)\tan \left( x + \frac{\pi}{4} \right) \\ & = \displaystyle \lim_{x \to \frac{\pi}{4} } \sin \left( \frac{\pi}{4} - x \right) . \frac{1}{\tan \left( \frac{\pi}{4} - x \right) } \\ & = \displaystyle \lim_{x \to \frac{\pi}{4} } \frac{\sin \left( \frac{\pi}{4} - x \right) }{\tan \left( \frac{\pi}{4} - x \right) } \\ & = 1 \end{align} $
Jadi, hasil limitnya adalah $ 1 . \, \heartsuit $
*). Mengubah bentuk $ \tan \left( x + \frac{\pi}{4} \right) $ :
$\begin{align} \tan \left( x + \frac{\pi}{4} \right) & = \frac{1}{\cot \left( x + \frac{\pi}{4} \right) } \\ & = \frac{1}{\tan \left( \frac{\pi}{2} - ( x + \frac{\pi}{4} ) \right) } \\ & = \frac{1}{\tan \left( \frac{\pi}{4} - x \right) } \end{align} $
*). Menyelesaikan soal :
$\begin{align} & \displaystyle \lim_{x \to \frac{\pi}{4} } \sin \left( \frac{\pi}{4} - x \right)\tan \left( x + \frac{\pi}{4} \right) \\ & = \displaystyle \lim_{x \to \frac{\pi}{4} } \sin \left( \frac{\pi}{4} - x \right) . \frac{1}{\tan \left( \frac{\pi}{4} - x \right) } \\ & = \displaystyle \lim_{x \to \frac{\pi}{4} } \frac{\sin \left( \frac{\pi}{4} - x \right) }{\tan \left( \frac{\pi}{4} - x \right) } \\ & = 1 \end{align} $
Jadi, hasil limitnya adalah $ 1 . \, \heartsuit $
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