Pembahasan Limit Trigonometri UM UGM 2005 Matipa kode 612

Soal yang Akan Dibahas
$ \displaystyle \lim_{x \to \frac{\pi}{4}} \frac{\left(x - \frac{\pi}{4} \right) \tan \left(3x - \frac{3\pi}{4} \right)}{2(1 - \sin 2x)} = .... $
A). $ 0 \, $ B). $ -\frac{3}{2} \, $ C). $ \frac{3}{2} \, $ D). $ -\frac{3}{4} \, $ E). $ \frac{3}{4} \, $

$\spadesuit $ Konsep Dasar
*). Sifat limit fungsi trigonometri :
$ \displaystyle \lim_{y \to 0} \frac{ay}{\sin by} = \frac{a}{b} $ dan $ \displaystyle \lim_{y \to 0} \frac{\tan ay}{\sin by} = \frac{a}{b} $
*). RUmus dasar trigonometri :
$ 1 - \cos 2A = 2 \sin A . \sin A $
$ \sin (\frac{\pi}{2} + A) = \cos A $

$\clubsuit $ Pembahasan
*). Misalkan $ y = x - \frac{\pi}{4} $ sehingga $ x = y + \frac{\pi}{4} $
untuk $ x $ mendekati $ \frac{\pi}{4} $ maka $ y $ mendekati nol.
*). Menyelesaikan soal
$ \begin{align} & \displaystyle \lim_{x \to \frac{\pi}{4}} \frac{\left(x - \frac{\pi}{4} \right) \tan \left(3x - \frac{3\pi}{4} \right)}{2(1 - \sin 2x)} \\ & = \displaystyle \lim_{x \to \frac{\pi}{4}} \frac{\left(x - \frac{\pi}{4} \right) \tan 3 \left(x - \frac{\pi}{4} \right)}{2(1 - \sin 2x)} \\ & = \displaystyle \lim_{y \to 0} \frac{y . \tan 3 y}{2(1 - \sin 2 \left(y + \frac{\pi}{4} \right) )} \\ & = \displaystyle \lim_{y \to 0} \frac{y . \tan 3 y}{2(1 - \sin \left(2y + \frac{\pi}{2} \right) )} \\ & = \displaystyle \lim_{y \to 0} \frac{y . \tan 3 y}{2(1 - \cos 2y )} \\ & = \displaystyle \lim_{y \to 0} \frac{y . \tan 3 y}{2( 2\sin y . \sin y) } \\ & = \displaystyle \lim_{y \to 0} \frac{1}{4} . \frac{y }{ \sin y } . \frac{\tan 3 y}{ \sin y } \\ & = \frac{1}{4} . \frac{1 }{ 1 } . \frac{3}{ 1 } = \frac{3}{4} \end{align} $
Jadi, hasil limitnya adalah $ \frac{3}{4} . \, \heartsuit $

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