Pembahasan Contoh Soal Materi Limit Fungsi Trigonometri #3

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

Contoh soal 1

Nilai dari $\lim\limits_{x \to 0} \dfrac{3x \tan 4x}{1- \cos^2 x}$ adalah ...

Jawab:

$\begin{aligned}  & \lim\limits_{x \to 0}  \dfrac{3x \tan 4x}{1- \cos^2 x} \\  & =  \lim\limits_{x \to 0}  \dfrac{3x \tan 4x}{\sin^2 x} \\ &=  \lim\limits_{x \to 0}  \dfrac{3x }{\sin x} \times \frac{\tan 4x}{\sin x} \\&= \frac{3}{1} \times \frac{4}{1}\\&= 12 \end{aligned}$

Contoh soal 2

Nilai dari $\lim\limits_{x \to 0} \dfrac{1- \cos^2 2x}{-x \sin2x}$ adalah ...

Jawab:

$\begin{aligned} &\lim\limits_{x \to 0} \dfrac{1- \cos^2 2x}{-x \sin2x} \\&= \lim\limits_{x \to 0} \dfrac{1- (1-\sin^2 2x)}{-x \sin2x}\\&= \lim\limits_{x \to 0} \dfrac{ \sin^2 2x}{-x \sin2x}\\&= \lim\limits_{x \to 0} \dfrac{\sin 2x}{-x }  \times \frac{\sin2x}{\sin2x}\\&= -2 \times 1\\&=-2 \end{aligned}$

Contoh soal 3

Nilai dari $\lim\limits_{x \to 0} \dfrac{1- \cos x}{4x \sin2x}$ adalah ...

Jawab:

$\begin{aligned} &\lim\limits_{x \to 0} \dfrac{1- \cos x}{4x \sin2x} \\&= \lim\limits_{x \to 0} \dfrac{1- (1-2 \sin^2 \frac{1}{2}x)}{4x \sin2x}\\&= \lim\limits_{x \to 0} \dfrac{2 \sin^2 \frac{1}{2}x}{4x \sin2x} \\&= \lim\limits_{x \to 0} 2 \dfrac{ \sin \frac{1}{2}x}{4x } \times \frac{\sin \frac{1}{2}x}{\sin2x} \\&= 2 \times \frac{1}{8} \times \frac{1}{4} \\&= \frac{1}{16}   \end{aligned}$

Contoh soal 4

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

Jawab:

Ingat rumus:

$\color{red}{\sin A + \sin B= 2 \sin \frac{1}{2}(A+B) \cos \frac{1}{2}(A-B)}$ dan

$\color{red}{\cos (-x)=cos x}$

$\begin{aligned} & \lim\limits_{x \to 0} \dfrac{ \sin x+\sin 3x}{x \cos x}\\&= \lim\limits_{x \to 0} \dfrac{ 2 \sin2x \cos (-x)}{x \cos x}\\&= \lim\limits_{x \to 0} 2 \dfrac{ \sin 2x}{x }\\&= 2 \times \frac{2}{1}\\&=4  \end{aligned}$

Contoh soal 5

Nilai dari $\lim\limits_{x \to 0} \dfrac{ 2x \tan 4x}{1- \cos^2 2x}$ adalah ...

Jawab:

Ingat rumus:

$\color{red}{1-\cos^2 A= \sin^2 A}$

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

Contoh soal 6

Nilai dari $\lim\limits_{x \to 0} \dfrac{ 1- \cos^2 x}{x^2 \cot(x+\frac{\pi}{4})}$ adalah ...

Jawab:

Ingat rumus:

$\color{red}{1-\cos^2 A= \sin^2 A}$

$\begin{aligned} &\lim\limits_{x \to 0} \dfrac{ 1- \cos^2 x}{x^2 \cot(x+\frac{\pi}{4})} \\&= \lim\limits_{x \to 0} \dfrac{  \sin^2 x}{x^2 \cot(x+\frac{\pi}{4})}\\&=  \lim\limits_{x \to 0} \dfrac{ \sin^2 x . \tan(x+\frac{\pi}{4})}{x^2 }  \\&= \lim\limits_{x \to 0} \dfrac{ \sin^2 x }{x^2 } \times  \tan(x+\frac{\pi}{4}) \\&= \left ( \frac{1}{1} \right )^2 \times \tan \left ( \frac{\pi}{4}  \right )\\&= 1 \times 1\\&= 1    \end{aligned}$

Contoh soal 7

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

Jawab:

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

Contoh soal 8

Nilai dari $\lim\limits_{x \to 0} \dfrac{ \cot 2x}{ \cot 4x}$ adalah ...

Jawab:

$\begin{aligned} &\lim\limits_{x \to 0} \dfrac{ \cot 2x}{ \cot 4x} \\&= \lim\limits_{x \to 0} \dfrac{ \frac{1}{\tan 2x}}{\frac{1}{ \tan 4x}}\\&= \lim\limits_{x \to 0} \dfrac{ \tan 4x}{ \tan 2x} \\&= \frac{4}{2}\\&=2    \end{aligned}$ 

Contoh soal 9

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

Jawab:

$\begin{aligned} &\lim\limits_{x \to 0} \dfrac{ 2x + 3x \cos 2x}{ \sin x \cos x}\\&= \lim\limits_{x \to 0} \dfrac{ 2x + 3x \cos 2x}{ \sin x \cos x} \times \frac{\frac{1}{x}}{\frac{1}{x}}\\&=  \lim\limits_{x \to 0} \dfrac{ \frac{2x}{x} + \frac{3x \cos 2x}{x}}{ \frac{\sin x \cos x}{x}}\\&= \lim\limits_{x \to 0} \dfrac{ 2 + 3 \cos 2x}{ \frac{\sin x}{x} \cos x} \\&= \frac{2 + 3 \cos 0}{1 \cos 0}\\&= \frac{5}{1} \\&=5  \end{aligned}$

Contoh soal 10

Nilai dari $\lim\limits_{x \to -3} \dfrac{ 1-\cos (x+3)}{(x+3)^2}$ adalah ...

Jawab:

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

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