Pembahasan Contoh Soal Materi Limit Fungsi Trigonometri #5
Berikut ini mimin sajikan beberapa contoh soal dan pembahasan pada materi limit fungsi trigonometri. Selamat membaca, sobat. Semoga bermanfaat.
Contoh soal 1
Nilai dari $\lim\limits_{x \to \frac{\pi}{4}} \dfrac{ \cos^2 x}{\tan x-\sin x}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to \frac{\pi}{4}} \dfrac{ \cos^2 x}{\tan x-\sin x}\\ &= \dfrac{ \cos^2 \frac{\pi}{4}}{\tan \frac{\pi}{4}-\sin \frac{\pi}{4}} \\ &= \dfrac{ (\frac{1}{2} \sqrt{2} )^2 }{1-\frac{1}{2} \sqrt{2}}\\ &= \dfrac{ \frac{1}{2}}{\frac{2- \sqrt{2}}{2}}\\ &= \frac{1}{2-\sqrt{2}}\\ &=\frac{2+\sqrt{2}}{2}\\ &= 1+\frac{1}{2} \sqrt{2} \end{aligned}$
Contoh soal 2
Nilai dari $\lim\limits_{x \to \frac{\pi}{3}} \dfrac{ \cos x-\sin 2x \cos x}{\cos^2 2x}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to \frac{\pi}{3}} \dfrac{ \cos x-\sin 2x \cos x}{\cos^2 2x}\\ &= \lim\limits_{x \to \frac{\pi}{3}} \dfrac{ \cos x(1-\sin 2x) }{1-\sin^2 2x}\\&= \lim\limits_{x \to \frac{\pi}{3}} \dfrac{ \cos x(1-\sin 2x) }{(1+\sin 2x)(1-\sin 2x)}\\ &= \lim\limits_{x \to \frac{\pi}{3}} \frac{\cos x}{1+\sin 2x}\\ &= \frac{\cos \frac{\pi}{3}}{1+\sin (2. \frac{\pi}{3})}\\&= \frac{\frac{1}{2}}{1+\frac{1}{2} \sqrt{3}}\\&= \frac{1}{2+\sqrt{3}} \\&= 2- \sqrt{3} \end{aligned}$
Contoh soal 3
Nilai dari $\lim\limits_{x \to 2} (\sin \dfrac{x^2-4}{x-2})$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 2} (\sin \dfrac{x^2-4}{x-2})\\&= \lim\limits_{x \to 2} (\sin \dfrac{(x-2)(x+2)}{x-2}) \\ &= \lim\limits_{x \to 2} \sin (x+2)\\ &= \sin (2+2)\\ &= \sin 4 \end{aligned}$
Contoh soal 4
Nilai dari $\lim\limits_{x \to \frac{\pi}{4} } \frac{\sec 2x+1}{\tan 2x}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to \frac{\pi}{4} } \frac{\sec 2x+1}{\tan 2x}\\ &= \lim\limits_{x \to \frac{\pi}{4} } \frac{\frac{1}{\cos 2x}+1}{\frac{\sin 2x}{\cos 2x}} \times \frac{\cos 2x}{\cos 2x} \\&= \lim\limits_{x \to \frac{\pi}{4} } \frac{1+\cos 2x}{\sin 2x}\\&= \lim\limits_{x \to \frac{\pi}{4} } \frac{1+(2 \cos^2x-1)}{2 \sin x \cos x}\\&= \lim\limits_{x \to \frac{\pi}{4} } \frac{2 \cos^2 x}{2 \sin x \cos x}\\ &= \lim\limits_{x \to \frac{\pi}{4}} \frac{\cos x}{\sin x}\\&= \frac{\cos \frac{\pi}{4}}{\sin \frac{\pi}{4}}\\&= \frac{\frac{1}{2} \sqrt{2}}{\frac{1}{2} \sqrt{2}}\\&= 1 \end{aligned}$
Contoh soal 5
Nilai dari $\lim\limits_{x \to 0} \frac{1-\cos^2 2x}{x^2 \tan(x+\frac{\pi}{4})}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 0} \frac{1-\cos^2 2x}{x^2 \tan(x+\frac{\pi}{4})}\\&= \lim\limits_{x \to 0} \frac{\sin^2 2x}{x^2 \tan(x+\frac{\pi}{4})}\\&= \lim\limits_{x \to 0} \frac{\sin 2x }{x} \times \frac{\sin 2x}{x} \times \frac{1}{\tan(x+\frac{\pi}{4})} \\ &= \frac{2}{1} \times \frac{2}{1} \times \frac{1}{1}\\&=4 \end{aligned}$
Contoh soal 6
Nilai dari $\lim\limits_{x \to 0} \frac{24 \tan^3 \frac{2}{3}x}{x \sin^2 6x}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 0} \frac{24 \tan^3 \frac{2}{3}x}{x \sin^2 6x}\\&= \lim\limits_{x \to 0} 24 \times \frac{\tan \frac{2}{3}x}{x} \times \frac{\tan \frac{2}{3}x}{\sin 6x} \times \frac{\tan \frac{2}{3}x}{\sin 6x}\\&= 24 \times \frac{\frac{2}{3}}{1} \times \frac{\frac{2}{3}}{6} \times \frac{\frac{2}{3}}{6} \\&= 24 \times \frac{2}{3} \times \frac{1}{9} \times \frac{1}{9} \\&= \frac{16}{81} \end{aligned}$
Contoh soal 7
Nilai dari $\lim\limits_{x \to 1} \frac{ \tan(x-1) \sin(1-\sqrt{x})}{x^2-2x+1}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 1} \frac{ \tan(x-1) \sin(1-\sqrt{x})}{x^2-2x+1}\\&= \lim\limits_{x \to 1} \frac{ \tan(x-1) \sin(1-\sqrt{x})}{(x-1)(\sqrt{x}-1)(\sqrt{x}+1)}\\&= \lim\limits_{(x-1) \to 0} \frac{\tan(x-1)}{(x-1)} \times (-1) \times \lim\limits_{(1-\sqrt{x}) \to 0} \frac{\sin (1-\sqrt{x})}{(1-\sqrt{x})} \times \lim\limits_{x \to 1} \frac{1}{\sqrt{x}+1}\\&= 1 \times (-1) \times 1 \times \frac{1}{2}\\&= -\frac{1}{2} \end{aligned}$
Contoh soal 8
Nilai dari $\lim\limits_{x \to \frac{\pi}{6}} \frac{\cot x}{\sin 2x+ \cos 2x}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to \frac{\pi}{6}} \frac{\cot x}{\sin 2x+ \cos 2x}\\&= \frac{\cot \frac{\pi}{6}}{\sin 2(\frac{\pi}{6})+ \cos 2(\frac{\pi}{6})}\\&= \frac{\sqrt{3}}{\frac{1}{2} \sqrt{3} + \frac{1}{2}}\\&= 3- \sqrt{3} \end{aligned}$
Contoh soal 9
Nilai dari $\lim\limits_{x \to 0} \frac{36 x^2 \sin \frac{4x}{3}}{\tan^3 \frac{2}{3}x}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 0} \frac{36 x^2 \sin \frac{4x}{3}}{\tan^3 \frac{2}{3}x}\\&= \lim\limits_{x \to 0} 36 \times \frac{x}{\tan \frac{2}{3}x} \times \frac{x}{\tan \frac{2}{3}x} \times \frac{\sin \frac{4x}{3}}{\tan \frac{2}{3}x}\\&= 36 \times \frac{1}{\frac{2}{3}} \times \frac{1}{\frac{2}{3}} \times \frac{\frac{4}{3}}{\frac{2}{3}}\\&= 36 \times \frac{3}{2} \times \frac{3}{2} \times 2 \\&=162 \end{aligned}$
Contoh soal 10
Nilai dari $\lim\limits_{x \to 0} \frac{(\frac{4x}{3})^3}{(\cos \frac{2x}{3} \tan \frac{2x}{3})^3}$ adalah ...
Jawab:
$\begin{aligned} &\lim\limits_{x \to 0} \frac{(\frac{4x}{3})^3}{(\cos \frac{2x}{3} \tan \frac{2x}{3})^3}\\&= \lim\limits_{x \to 0} \frac{(\frac{4x}{3})^3}{(\cos \frac{2x}{3} \frac{\sin \frac{2x}{3}}{\cos \frac{2x}{3}})^3}\\&= \lim\limits_{x \to 0} \frac{(\frac{4x}{3})^3}{(\sin \frac{2x}{3} )^3}\\&= \frac{\frac{4}{3}}{\frac{2}{3}} \times \frac{\frac{4}{3}}{\frac{2}{3}} \times \frac{\frac{4}{3}}{\frac{2}{3}}\\&= 8 \end{aligned}$
Demikianlah beberapa contoh soal dan pembahasan pada materi limit fungsi trigonomteri. Semoga bermanfaat.
