PAS MATEMATIKA BLOG: Contoh Soal Lanjutan 3 Limit Fungsi Trigonometri (Matematika Peminatan Kelas XII)

$\begin{array}{ll}\\ 16.&\textrm{Nilai dari}\: \: \underset{x\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \cos 4x=....\\ &\begin{array}{ll}\\ \textrm{a}.&\displaystyle -1\\\\ \textrm{b}.&-\displaystyle \frac{1}{2}\sqrt{3}\\\\ \textrm{c}.&\displaystyle -\frac{1}{2}\\\\ \textrm{d}.&\displaystyle 0\\\\ \textrm{e}.&\displaystyle 1 \end{array} \end{array}$

Jawab : e

Cukup jelas

Dengan substitusi langsung kita akan dapat mentukan hasilnya, yatu:

$\color{blue}\begin{aligned}\underset{x\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \cos 4x=\cos 4(0)=\cos 0=1 \end{aligned}$

$\begin{array}{ll}\\ 17.&\textrm{Diketahui}\: \: f(x)=\begin{cases} \sin 2x, & \text{ saat } x< 0 \\ \tan x, & \text{ saat } x\geq 0 \end{cases}\\\\ &\textrm{Nilai dari}\: \: \underset{x\rightarrow \displaystyle 0^{+}}{\textrm{Lim}}\: \: f(x)=....\\ &\begin{array}{ll}\\ \textrm{a}.&\displaystyle -2\\\\ \textrm{b}.&-\displaystyle 1\\\\ \textrm{c}.&\displaystyle 0\\\\ \textrm{d}.&\displaystyle 1\\\\ \textrm{e}.&\displaystyle \textrm{tidak ada} \end{array} \end{array}$

Jawab : c

Berikut penjelasannya

$\color{blue}\begin{aligned}\underset{x\rightarrow \displaystyle 0^{+}}{\textrm{Lim}}\: \: \displaystyle f(x)&=\underset{x\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \tan x=\tan 0=0 \end{aligned}$

$\begin{array}{ll}\\ 17.&\textrm{Nilai dari}\: \: \underset{x\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{1-\cos 2x}{5\sin ^{2}x}=....\\ &\begin{array}{ll}\\ \textrm{a}.&\displaystyle -\frac{2}{5}\\\\ \textrm{b}.&\displaystyle \frac{2}{5}\\\\ \textrm{c}.&\displaystyle 1\\\\ \textrm{d}.&\displaystyle 2\\\\ \textrm{e}.&\displaystyle \infty \end{array} \end{array}$

Jawab : b

$\color{blue}\begin{aligned}\underset{x\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{ 1-\cos 2x}{5\sin ^{2}x}&=\underset{x\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{ 2\sin ^{2}x}{5\sin ^{2}x}\\ &=\underset{x\rightarrow \displaystyle 0}{\textrm{Lim}}\: \: \displaystyle \frac{2\color{red}\sin \textcircled{1}x\times \sin \textcircled{1}x}{5\sin \color{red}\textcircled{1}x\times \sin \textcircled{1}x}\\ &=\displaystyle \frac{2\times 1\times 1}{5\times 1\times 1}\\ &=\displaystyle \frac{2}{5}\\ &\\ &\\ \color{black}\textrm{Perlu diingat}\: &\textrm{bahwa}\quad 1-\cos 2x=2\sin ^{2}x \end{aligned}$

$\begin{array}{ll}\\ 18.&\textrm{Nilai dari}\: \: \underset{x\rightarrow \displaystyle \frac{\pi }{2}}{\textrm{Lim}}\: \: \displaystyle \frac{3-\sin x}{\cos 2x}=....\\ &\begin{array}{ll}\\ \textrm{a}.&\displaystyle -4\\\\ \textrm{b}.&\displaystyle -2\\\\ \textrm{c}.&\displaystyle \frac{1}{2}\\\\ \textrm{d}.&\displaystyle 1\\\\ \textrm{e}.&\displaystyle 2 \end{array} \end{array}$

Jawab : b

$\begin{array}{ll}\\ 19.&\textrm{Nilai dari}\: \: \underset{x\rightarrow \displaystyle \pi }{\textrm{Lim}}\: \: \displaystyle \frac{\sin x-2}{\cos x-3}=....\\ &\begin{array}{ll}\\ \textrm{a}.&\displaystyle \frac{1}{2}\\\\ \textrm{b}.&\displaystyle -1\\\\ \textrm{c}.&\displaystyle \frac{1}{3}\\\\ \textrm{d}.&\displaystyle 1\\\\ \textrm{e}.&\displaystyle 2 \end{array} \end{array}$

Jawab : a

$\color{blue}\begin{aligned}\underset{x\rightarrow \displaystyle \pi }{\textrm{Lim}}\: \: \displaystyle \frac{ \sin x-2}{\cos x-3}&\qquad\color{black}\textrm{dengan substitusi langsung didapatkan}\\ &=\underset{x\rightarrow \displaystyle \pi }{\textrm{Lim}}\: \: \displaystyle \frac{ \left (\sin \pi \right ) -2}{\left (\cos \pi \right ) -3}\\ &=\underset{x\rightarrow \displaystyle \frac{\pi }{2}}{\textrm{Lim}}\: \: \displaystyle \frac{\color{red}0-2}{-1-3}\\ &=\displaystyle \frac{-2}{-4}\\ &=\displaystyle \frac{1}{2}\\ \end{aligned}$

$\begin{array}{ll}\\ 20.&\textrm{Nilai dari}\: \: \underset{x\rightarrow \displaystyle \frac{\pi }{6}}{\textrm{Lim}}\: \: \displaystyle \left (\cos ^{2}x-\sin ^{2}x \right )=....\\ &\begin{array}{ll}\\ \textrm{a}.&\displaystyle -\frac{1}{2}\\\\ \textrm{b}.&\displaystyle \frac{1}{2}\\\\ \textrm{c}.&\displaystyle \frac{1}{3}\\\\ \textrm{d}.&\displaystyle 1\\\\ \textrm{e}.&\displaystyle 2 \end{array} \end{array}$

Jawab : b

$\color{blue}\begin{aligned}\underset{x\rightarrow \displaystyle \frac{\pi }{6}}{\textrm{Lim}}\: \: \displaystyle \left ( \cos ^{2}x-\sin ^{2}x \right )&\qquad \color{black}\textrm{dengan substitusi langsung}\\ &=\cos ^{2}\left ( \displaystyle \frac{\pi }{6} \right )-\sin ^{2}\left ( \displaystyle \frac{\pi }{6} \right )\\ &=\left ( \displaystyle \frac{1}{2}\sqrt{3} \right )^{2}-\left ( \displaystyle \frac{1}{2} \right )^{2}\\ &=\displaystyle \frac{3}{4}-\frac{1}{4}=\frac{2}{4}\\ &=\displaystyle \frac{1}{2} \end{aligned}$

Daftar Pustaka

1. Tim. 2020. Modul Matematika (Peminatan Matematika dan Ilmu-Ilmu Alam Kelas XII. Tangerang: RAHMA GEMILANG.

PAS MATEMATIKA BLOG

Contoh Soal Lanjutan 3 Limit Fungsi Trigonometri (Matematika Peminatan Kelas XII)

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