PAS MATEMATIKA BLOG: Contoh Soal Lanjutan 4 Limit Fungsi Trigonometri (Matematika Peminatan Kelas XII)

$\begin{array}{ll}\\ 21.&\textrm{Nilai dari}\: \: \underset{x\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{x\tan 3x}{1-\cos ^{2}2x}=....\\ &\begin{array}{ll}\\ \textrm{a}.&\displaystyle -\frac{3}{2}\\\\ \textrm{b}.&\displaystyle -\frac{3}{4}\\\\ \textrm{c}.&\displaystyle -\frac{1}{2}\\\\ \textrm{d}.&\displaystyle \frac{1}{2}\\\\ \textrm{e}.&\displaystyle \frac{3}{4} \end{array} \end{array}$

Jawab : e

Berikut pembahasannya

$\color{blue}\begin{aligned}\underset{x\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{x\tan 3x}{1-\cos ^{2}2x}&=\underset{x\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{x\tan 3x}{\sin ^{2}2x}\\ &=\underset{x\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{x\times \tan \textcircled{3}x}{\sin \textcircled{2}x\times \sin \textcircled{2}x}\\ &=\displaystyle \frac{1\times 3}{2\times 2}\\ &=\displaystyle \frac{3}{4} \end{aligned}$

$\begin{array}{ll}\\ 22.&\textrm{Nilai dari}\: \: \underset{x\rightarrow \displaystyle 3}{\textrm{Lim}}\: \: \displaystyle \frac{x^{2}-9}{\sin (x-3)}=....\\ &\begin{array}{ll}\\ \textrm{a}.&\displaystyle 9\\\\ \textrm{b}.&\displaystyle 7\\\\ \textrm{c}.&\displaystyle 6\\\\ \textrm{d}.&\displaystyle 3\\\\ \textrm{e}.&\displaystyle -6 \end{array} \end{array}$

Jawab : c

$\color{blue}\begin{aligned}\underset{x\rightarrow \displaystyle 3}{\textrm{Lim}}\: \: \displaystyle \frac{x^{2}-9}{\sin (x-3)}&\quad \textrm{perhatikan bahwa}\\ &\color{black}\textrm{misal}\: \: x=3+h,\: \: \textrm{maka}\: \: h\rightarrow 0\\ &\color{red}\textrm{selanjutnya gunakan}\: \: h=x-3,\: \: \textrm{dengan}\: \: h\rightarrow 0\\ &=\underset{h\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{(3+h)^{2}-9}{\sin h}\\ &=\underset{h\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{h^{2}+6h+9-9}{\sin h}\\ &=\underset{h\rightarrow \displaystyle 0}{\textrm{Lim}}\: \:\frac{h(h+6)}{\sin h}\\ &=\underset{h\rightarrow \displaystyle 0}{\textrm{Lim}}\: \:(h+6)\\ &=0+6=6 \end{aligned}$

atau boleh juga

$\color{blue}\begin{aligned}\underset{x\rightarrow \displaystyle 3}{\textrm{Lim}}\: \: \displaystyle \frac{x^{2}-9}{\sin (x-3)}&=\underset{x\rightarrow \displaystyle 3}{\textrm{Lim}}\: \: \displaystyle \frac{(x+3)(x-3)}{\sin (x-3)}\\ &=\underset{x\rightarrow \displaystyle 3}{\textrm{Lim}}\: \: \displaystyle (x+3)\\ &=3+3=6 \end{aligned}$

$\begin{array}{ll}\\ 23.&\textrm{Nilai dari}\: \: \underset{x\rightarrow \displaystyle \frac{\pi }{2}}{\textrm{Lim}}\: \: \displaystyle \frac{\sin 6x}{x-\displaystyle \frac{\pi }{2}}=....\\ &\begin{array}{ll}\\ \textrm{a}.&\displaystyle 6\\\\ \textrm{b}.&\displaystyle 3\\\\ \textrm{c}.&\displaystyle 2\\\\ \textrm{d}.&\displaystyle -2\\\\ \textrm{e}.&\displaystyle -6 \end{array} \end{array}$

Jawab : e

$\color{blue}\begin{aligned}\underset{x\rightarrow \displaystyle \frac{\pi }{2}}{\textrm{Lim}}\: \: \displaystyle \frac{\sin 6x}{x-\displaystyle \frac{\pi }{2}}&\quad \color{black}\textrm{perhatikan juga bahwa}\\ &x=\displaystyle \frac{\pi }{2}+h,\quad \textrm{dengan}\: \: \: h\rightarrow 0\\ &\color{red}\textrm{atau}\: \: \: h=x-\displaystyle \frac{\pi }{2}\quad \textrm{dengan}\: \: \: h\rightarrow 0\\ &=\underset{h\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{\sin 6\left ( \displaystyle \frac{\pi }{2}+h \right )}{h}=\underset{h\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{\sin \left ( \displaystyle 3\pi +6h \right )}{h}\\ &=\underset{h\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{\sin \left ( \displaystyle \pi +6h \right )}{h}=\underset{h\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{-\sin \left ( \displaystyle 6h \right )}{h}\\ &=-6 \end{aligned}$

Daftar Pustaka

1. Astuti, A. N., Miyanto, dan Ngapiningsih. 2020. Matematika untuk SMA/MA Peminatan Matematika dan Ilmu-Ilmu Alam Kelas XII. Yogyakarta: PT. PENERBIT INTAN PARIWARA.

PAS MATEMATIKA BLOG

Contoh Soal Lanjutan 4 Limit Fungsi Trigonometri (Matematika Peminatan Kelas XII)

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