2010 Pembahasan Limit Trigonometri UTUL UGM Matematika Ipa

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 $

$\spadesuit $ Konsep Dasar
*). Penerapan Turunan pada Limit :
Jika $ \displaystyle \lim_{x \to k} \frac{f(x)}{g(x)} = \frac{0}{0} $ , maka solusinya $ \displaystyle \lim_{x \to k} \frac{f(x)}{g(x)} = \displaystyle \lim_{x \to k} \frac{f^\prime (x)}{g^\prime (x)} $ .
*). Turunan fungsi trigonometri :
$ y = \sin f(x) \rightarrow y^\prime = f^\prime (x) \cos f(x) $
$ y = \cos f(x) \rightarrow y^\prime = -f^\prime (x) \sin f(x) $
*). Rumus dasar trigonometri :
$ \tan f(x) = \frac{\sin f(x)}{\cos f(x) } $

$\clubsuit $ Pembahasan
*). Menentukan turunan trigonometrinya :
$ y = \sin \left( \frac{\pi}{4} - x \right) \rightarrow y^\prime = - \cos \left( \frac{\pi}{4} - x \right) $
$ y = \cos \left( x + \frac{\pi}{4} \right) \rightarrow y^\prime = - \sin \left( x + \frac{\pi}{4} \right) $
*). 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{\sin \left( x + \frac{\pi}{4} \right) }{ \cos \left( x + \frac{\pi}{4} \right) } \\ & = \displaystyle \lim_{x \to \frac{\pi}{4} } \sin \left( x + \frac{\pi}{4} \right) \frac{\sin \left( \frac{\pi}{4} - x \right) }{ \cos \left( x + \frac{\pi}{4} \right) } \\ & = \displaystyle \lim_{x \to \frac{\pi}{4} } \sin \left( x + \frac{\pi}{4} \right) \times \displaystyle \lim_{x \to \frac{\pi}{4} } \frac{\sin \left( \frac{\pi}{4} - x \right) }{ \cos \left( x + \frac{\pi}{4} \right) } \, \, \, \, \, \, \text{(turunan)} \\ & = \displaystyle \lim_{x \to \frac{\pi}{4} } \sin \left( x + \frac{\pi}{4} \right) \times \displaystyle \lim_{x \to \frac{\pi}{4} } \frac{- \cos \left( \frac{\pi}{4} - x \right)}{ - \sin \left( x + \frac{\pi}{4} \right) } \\ & = \sin \left( \frac{\pi}{4} + \frac{\pi}{4} \right) \times \frac{ \cos \left( \frac{\pi}{4} - \frac{\pi}{4} \right)}{ \sin \left( \frac{\pi}{4} + \frac{\pi}{4} \right) } \\ & = \sin \left( \frac{\pi}{2} \right) \times \frac{ \cos \left( 0 \right)}{ \sin \left( \frac{\pi}{2} \right) } \\ & = 1 \times \frac{ 1}{ 1 } = 1 \end{align} $
Jadi, hasil limitnya adalah $ 1 . \, \heartsuit $



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