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

$\begin{array}{ll}\\ 1.&\textrm{Perhatikanlah pernyataan-pernyataan berikut}\\ &(\textrm{i})\quad\textrm{Jika}\: \: \underset{x\rightarrow 0^{-} }{\textrm{Lim}}\: \: f(x)=4\: \: \textrm{dan}\: \: \textrm{Jika}\: \: \underset{x\rightarrow 0^{+} }{\textrm{Lim}}\: \: f(x)=2,\\ &\qquad\textrm{maka nilai}\: \: \underset{x\rightarrow 0 }{\textrm{Lim}}\: \: f(x)=8\\ &(\textrm{ii})\: \: \: \textrm{Jika}\: \: \underset{x\rightarrow 0^{-} }{\textrm{Lim}}\: \: f(x)=4\: \: \textrm{dan}\: \: \textrm{Jika}\: \: \underset{x\rightarrow 0^{+} }{\textrm{Lim}}\: \: f(x)=4,\\ &\qquad\textrm{maka nilai}\: \: \underset{x\rightarrow 0 }{\textrm{Lim}}\: \: f(x)=4\\ &(\textrm{iii})\: \: \textrm{Jika}\: \: \underset{x\rightarrow 0^{-} }{\textrm{Lim}}\: \: f(x)=4\: \: \textrm{dan}\: \: \textrm{Jika}\: \: \underset{x\rightarrow 0^{+} }{\textrm{Lim}}\: \: f(x)=2,\\ &\qquad\textrm{maka nilai}\: \: \underset{x\rightarrow 0 }{\textrm{Lim}}\: \: f(x)\: \: \textrm{tidak ada}\\ &(\textrm{iv})\: \: \textrm{Jika}\: \: \underset{x\rightarrow 0^{-} }{\textrm{Lim}}\: \: f(x)=3\: \: \textrm{dan}\: \: \textrm{Jika}\: \: \underset{x\rightarrow 0^{+} }{\textrm{Lim}}\: \: f(x)=2,\\ &\qquad\textrm{maka nilai}\: \: \underset{x\rightarrow 0 }{\textrm{Lim}}\: \: f(x)=1\\ \end{array}$

Pernyataan di atas yang tepat adalah ...

$\begin{array}{ll}\\ \textrm{a}.&(\textrm{i})\: \: \textrm{dan}\: \: (\textrm{ii})\\ \textrm{b}.&(\textrm{i})\: \: \textrm{dan}\: \: (\textrm{iii})\\ \textrm{c}.&(\textrm{ii})\: \: \textrm{dan}\: \: (\textrm{iii})\\ \textrm{d}.&(\textrm{ii})\: \: \textrm{dan}\: \: (\textrm{iv})\\ \textrm{e}.&(\textrm{iii})\: \: \textrm{dan}\: \: (\textrm{iv}) \end{array}$

Jawab : c

Perhatikan definisi limit di sini

$\begin{array}{ll}\\ 2.&\textrm{Perhatikan gambar dan pernyataan-pernyataan berikut}\\ \end{array}$

$\begin{array}{ll}\\ &(\textrm{i})\quad\textrm{Nilai}\: \: \underset{x\rightarrow \displaystyle \frac{\pi }{2} }{\textrm{Lim}}\: \: f(x)=4\\ &(\textrm{ii})\: \: \: \textrm{Nilai}\: \: \underset{x\rightarrow \displaystyle \frac{3\pi }{4} }{\textrm{Lim}}\: \: f(x)=\sqrt{2}\\ &(\textrm{iii})\: \: \textrm{Nilai}\: \: \underset{x\rightarrow \displaystyle \pi }{\textrm{Lim}}\: \: f(x)=1\\\\ &(\textrm{iv})\: \: \textrm{Nilai}\: \: \underset{x\rightarrow \displaystyle \frac{5\pi }{4} }{\textrm{Lim}}\: \: f(x)=-1\\ &\textrm{Pernyataan-pernyataan yang tepat ditunjukkan oleh}....\\ \end{array}$

Jawab : a

$\begin{array}{ll}\\ 3.&\textrm{Perhatikan tabel nilai-nilai}\: \: f(x)\: \: \textrm{berikut}\\ \end{array}$

$\begin{array}{|c|c|}\hline x&f(x)\\\hline -1&2,37679\\\hline -0,2&2,01339\\\hline -0,1&2,00334\\\hline -0,01&2,00003\\\hline \cdots &\cdots \\\hline 0&?\\\hline \cdots &\cdots \\\hline 0,01&2,00003\\\hline 0,1&2,00334\\\hline 0,2&2,01339\\\hline 1&2,37679\\\hline \end{array}$

dan

(i) $\underset{x\rightarrow 0 }{\textrm{lim}}\: f(x)=0$

(ii) $\underset{x\rightarrow 0 }{\textrm{lim}}\: f(x)=1$

(iii) $\underset{x\rightarrow 0^{-} }{\textrm{lim}}\: f(x)=2$

(iv) $\underset{x\rightarrow 0^{+} }{\textrm{lim}}\: f(x)=2$

Dari pernyataan yang sesuai tabel di atas

yang tepat adalah ...

Jawab : e

Cukup jelas baik dengan limit kiri dan limit kanan diperoleh nilai limitnya = 2 saat x mendekati nol.

$\begin{array}{ll}\\ 4.&\textrm{Nilai dari}\: \: \underset{x\rightarrow \displaystyle \frac{\pi }{2} }{\textrm{Lim}}\: \: \cos x=....\\ \end{array}$

$\begin{array}{ll}\\ \textrm{a}.&-1\\\\ \textrm{b}.&-\displaystyle \frac{1}{2}\\\\ \textrm{c}.&0\\\\ \textrm{d}.&\displaystyle \frac{1}{2}\\\\ \textrm{e}.&1 \end{array}$

Jawab : c

$\color{blue}\begin{aligned}\textrm{Kita Selesai}&\textrm{kan dengan substitusi langsung, yaitu:}\\ \underset{x\rightarrow \displaystyle \frac{\pi }{2} }{\textrm{Lim}}\: \: \cos x&=\cos \displaystyle \frac{\pi }{2}=\cos 90^{\circ}=0 \end{aligned}$

$\begin{array}{ll}\\ 5.&\textrm{Nilai dari}\: \: \underset{x\rightarrow \displaystyle \frac{\pi }{4} }{\textrm{Lim}}\: \: \sin \left ( 2x-\displaystyle \frac{\pi }{6} \right )=....\\ \end{array}$

$\begin{array}{ll}\\ \textrm{a}.&-1\\\\ \textrm{b}.&0\\\\ \textrm{c}.&\displaystyle \frac{1}{2}\\\\ \textrm{d}.&\displaystyle \frac{1}{2}\sqrt{2}\\\\ \textrm{e}.&\displaystyle \frac{1}{2}\sqrt{3} \end{array}$

Jawab : e

Berikut pembahasannya

$\color{blue}\begin{aligned}\textrm{Kita Selesaikan deng}&\textrm{an substitusi langsung, yaitu:}\\ \underset{x\rightarrow \displaystyle \frac{\pi }{4} }{\textrm{Lim}}\: \: \sin \left ( 2x-\displaystyle \frac{\pi }{6} \right )&=\sin \left ( 2.\displaystyle \frac{\pi }{4}-\frac{\pi }{6} \right )\\ &=\sin \left ( \displaystyle \frac{\pi }{2}-\frac{\pi }{6} \right )\\ &=\sin \left ( 90^{\circ}-30^{\circ} \right )\\ &=\sin 60^{\circ}\\ &=\displaystyle \frac{1}{2}\sqrt{3} \end{aligned}$

PAS MATEMATIKA BLOG

Contoh Soal Limit Fungsi Trigonometri (Matematika Peminatan Kelas XII)

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