Pembahasan Contoh Soal Materi Limit Fungsi Trigonometri #2

 

Berikut ini mimin sajikan beberapa contoh soal dan pembahasan pada materi limit fungsi trigonometri.  Selamat membaca, sobat. Semoga bermanfaat. 

Contoh soal 1

Nilai dari $\lim\limits_{x \to 0} \dfrac{\sin 5x+\sin 4x}{ \sin 3x}$ adalah ...

Jawab:

$\begin{aligned}  & \lim\limits_{x \to 0}  \dfrac{\sin 5x+\sin 4x}{ \sin 3x} \\  & =  \lim\limits_{x \to 0}  \dfrac{\sin 5x}{ \sin 3x} +  \dfrac{\sin 4x}{ \sin 3x} \\ &= \frac{5}{3} +\frac{4}{3}\\&=\frac{9}{3}\\&=3  \end{aligned}$

Contoh soal 2

Nilai dari $\lim\limits_{x \to 0} \dfrac{1-\cos^{2} x}{1- \cos x}$ adalah ...

Jawab:

$\begin{aligned}  & \lim\limits_{x \to 0}  \dfrac{1-\cos^{2} x}{1- \cos x} \\  & =  \lim\limits_{x \to 0}  \dfrac{(1- \cos x)(1+ \cos x)}{1- \cos x} \\ &=  \lim\limits_{x \to 0}  1+ \cos x \\&=1+1 \\&=2  \end{aligned}$

Contoh soal 3

Nilai dari $\lim\limits_{x \to 0} \sqrt{\dfrac{1-\cos x}{x^2}}$ adalah ...

Jawab:

$\begin{aligned}  & \lim\limits_{x \to 0}  \sqrt{\dfrac{1-\cos x}{x^2}}  \\ &=  \sqrt{ \lim\limits_{x \to 0} \dfrac{1-(1-2\sin^{2} \frac{1}{2}x)}{x^2}}\\&=  \sqrt{\lim\limits_{x \to 0} \dfrac{1-1+2\sin^{2} \frac{1}{2}x)}{x^2}}\\&=  \sqrt{\lim\limits_{x \to 0} 2 \times \dfrac{\sin \frac{1}{2}x}{x} \times \dfrac{\sin \frac{1}{2}x}{x}}\\&= \sqrt{2 \times \frac{1}{2} \times \frac{1}{2}} \\&= \sqrt{\frac{2}{4}}\\&= \frac{1}{2} \sqrt{2} \end{aligned}$

Contoh soal 4

Nilai dari $\lim\limits_{x \to 0} \dfrac{\cos x-\cos^{3} x}{x^2}$ adalah ...

Jawab:

$\begin{aligned} &\lim\limits_{x \to 0} \dfrac{\cos x-\cos^{3} x}{x^2} \\&= \lim\limits_{x \to 0} \dfrac{\cos x(1-\cos^{2} x)}{x^2}\\&= \lim\limits_{x \to 0} \dfrac{\cos x \sin^{2} x}{x^2} \\ &= \lim\limits_{x \to 0} \cos x \times \dfrac{\sin  x}{x} \times \dfrac{\sin  x}{x}\\&= 1 \times 1 \times 1\\&=1  \end{aligned}$

Contoh soal 5

Nilai dari $\lim\limits_{x \to 0} x \cot 2x$ adalah ...

Jawab:

$\begin{aligned} &\lim\limits_{x \to 0} x \cot 2x  \\&= \lim\limits_{x \to 0} x \times \frac{\cos 2x}{\sin 2x}\\&= \lim\limits_{x \to 0}  \frac{x}{\sin 2x} \times \cos 2x \\&= \frac{1}{2} \times 1 \\&= \frac{1}{2} \end{aligned}$

Contoh soal 6

Nilai dari $\lim\limits_{x \to 0} \frac{\cos 2x \sin 5x}{8x}$ adalah ...

Jawab:

$\begin{aligned} &\lim\limits_{x \to 0} \frac{\cos 2x \sin 5x}{8x}  \\&= \lim\limits_{x \to 0} \cos 2x \times  \frac{\sin 5x}{8x} \\&= 1 \times \frac{5}{8}\\&= \frac{5}{8}   \end{aligned}$

Contoh soal 7

Nilai dari $\lim\limits_{x \to \frac{\pi}{3}} \frac{\tan (3x-\pi) \cos 2x}{\sin (3x-\pi)}$ adalah ...

Jawab:

$\begin{aligned} &\lim\limits_{x \to \frac{\pi}{3}} \frac{\tan (3x-\pi) \cos 2x}{\sin (3x-\pi)}\\&= \lim\limits_{x \to \frac{\pi}{3}} \frac{\tan (3x-\pi) }{\sin (3x-\pi)} \times \cos 2x\\&=  \lim\limits_{x \to \frac{\pi}{3}} \cos 2x\\&= \cos (\frac{2 \pi}{3}) \\&= -\frac{1}{2}  \end{aligned}$

Contoh soal 8

Nilai dari $\lim\limits_{x \to 0} {\dfrac{1-\cos 2x}{x^2}}$ adalah ...

Jawab:

$\begin{aligned} &\lim\limits_{x \to 0} {\dfrac{1-\cos 2x}{x^2}}\\&= \lim\limits_{x \to 0} {\dfrac{1-(1-2 \sin^{2}x)}{x^2}}\\&= \lim\limits_{x \to 0} {\dfrac{2 \sin^{2}x}{x^2}}  \\&= \lim\limits_{x \to 0} 2 \left ( \frac{\sin x}{x} \right )^{2}\\&= 2.1^2\\&=2    \end{aligned}$

Contoh soal 9

Nilai dari $\lim\limits_{x \to 0} {\dfrac{1-\cos 4x}{1-\cos 6x}}$ adalah ...

Jawab:

$\color{red}{\text{Noted:   } 1- \cos kx=2 \sin^2 \frac{1}{2}kx }$

$\begin{aligned} &\lim\limits_{x \to 0} {\dfrac{1-\cos 4x}{1-\cos 6x}} \\&= \lim\limits_{x \to 0} {\dfrac{2 \sin^2 2x}{2 \sin^2 3x}} \\&= \left ( \lim\limits_{x \to 0} {\dfrac{ \sin 2x}{ \sin 3x}}  \right )^2 \\&= \left (\frac{2}{3}  \right )^2\\&=\frac{4}{9}  \end{aligned}$

Contoh soal 10

Nilai dari $\lim\limits_{x \to 0} \frac{\sqrt{9x}}{\sqrt{\sin 3x}}$ adalah ...

Jawab:

$\begin{aligned} &\lim\limits_{x \to 0} \frac{\sqrt{9x}}{\sqrt{\sin 3x}}\\&= \sqrt{\lim\limits_{x \to 0} \frac{9x}{\sin 3x}}\\&= \sqrt{\frac{9}{3}}\\&=\sqrt{3}  \end{aligned}$

Demikianlah beberapa contoh soal dan pembahasan pada materi limit fungsi trigonomteri. Semoga bermanfaat.