Pembahasan Contoh Soal Materi Limit Fungsi Trigonometri

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

Sebagai pengingat berikut ini teorema dasar limit fungsi trigonometri.

Rumus-rumus pendukung (identitas trigonometri dasar)

Contoh soal 1

Nilai dari $\underset{x\to 0}{lim}{\displaystyle \frac{\sin 8x}{8x}}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 0}{lim}{\displaystyle \frac{\sin 8x}{8x}}\\ & ={\displaystyle \frac{8}{8}}\\ & =1\end{array}$

Contoh soal 2

Nilai dari $\underset{x\to 0}{lim}{\displaystyle \frac{\sin 5x}{3x}}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 0}{lim}{\displaystyle \frac{\sin 5x}{3x}}\\ & ={\displaystyle \frac{5}{3}}\end{array}$

Contoh soal 3

Nilai dari $\underset{x\to 0}{lim}{\displaystyle \frac{\tan 6x}{x}}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 0}{lim}{\displaystyle \frac{\tan 6x}{x}}\\ & ={\displaystyle \frac{6}{1}}\\ & =6\end{array}$

Contoh soal 4

Nilai dari $\underset{x\to 0}{lim}{\displaystyle \frac{\sin 5x}{\sin 7x}}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 0}{lim}{\displaystyle \frac{\sin 5x}{\sin 7x}}\\ & ={\displaystyle \frac{5}{7}}\end{array}$

Contoh soal 5

Nilai dari $\underset{x\to 0}{lim}{\displaystyle \frac{\tan 6x}{\sin 3x}}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 0}{lim}{\displaystyle \frac{\tan 6x}{\sin 3x}}\\ & ={\displaystyle \frac{6}{3}}\\ & =2\end{array}$

Contoh soal 6

Nilai dari $\underset{x\to 0}{lim}{\displaystyle \frac{\tan 5x}{\tan 4x}}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 0}{lim}{\displaystyle \frac{\tan 5x}{\tan 4x}}\\ & ={\displaystyle \frac{5}{4}}\end{array}$

Contoh soal 7

Nilai dari $\underset{x\to 0}{lim}{\displaystyle \frac{\tan 2x}{3x}}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 0}{lim}{\displaystyle \frac{\tan 2x}{3x}}\\ & ={\displaystyle \frac{2}{3}}\end{array}$

Contoh soal 8

Nilai dari $\underset{x\to 0}{lim}{\displaystyle \frac{x^2}{1-\cos x}}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 0}{lim}{\displaystyle \frac{x^2}{1-\cos x}}\times \frac{1+\cos x}{1+\cos x}\\ & =\underset{x\to 0}{lim}{\displaystyle \frac{x^2(1+\cos x)}{1-{\cos }^{2}x}}\\ & =\underset{x\to 0}{lim}{\displaystyle \frac{x^2(1+\cos x)}{{\sin }^{2}x}}\\ & =\underset{x\to 0}{lim}{\displaystyle \frac{x}{\sin x}}\times \frac{x}{\sin x}\times \frac{1+\cos x}{1}\\ & =\frac{1}{1}\times \frac{1}{1}\times \frac{2}{1}\\ & =2\end{array}$

Contoh soal 9

Nilai dari $\underset{x\to 0}{lim}{\displaystyle \frac{{\sin }^2\frac{1}{2}x}{x^2}}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 0}{lim}{\displaystyle \frac{{\sin }^2\frac{1}{2}x}{x^2}}\\ & =\underset{x\to 0}{lim}{\displaystyle \frac{\sin \frac{1}{2}x}{x}}\times {\displaystyle \frac{\sin \frac{1}{2}x}{x}}\\ & =\frac{\frac{1}{2}}{1}\times \frac{\frac{1}{2}}{1}\\ & =\frac{1}{4}\end{array}$

Contoh soal 10

Nilai dari $\underset{x\to 0}{lim}{\displaystyle \frac{1-\cos 2x}{x\sin x}}$ adalah ...

Jawab:

$\begin{array}{rl}& \underset{x\to 0}{lim}{\displaystyle \frac{1-\cos 2x}{x\sin x}}\\ & =\underset{x\to 0}{lim}{\displaystyle \frac{2{\sin }^{2}x}{x\sin x}}\\ & =\underset{x\to 0}{lim}2\times {\displaystyle \frac{\sin x}{x}}\times \frac{\sin x}{\sin x}\\ & =2\times \frac{1}{1}\times \frac{1}{1}\\ & =2\end{array}$

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