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Selamat datang di halaman contoh soal dan pembahasan "Limit Trigonometri".
Di halaman ini akan membahas tentang contoh soal dan pembahasan lengkap mengenai Limit Trigonometri.
Untuk lebih jelasnya mari kita lihat contoh soal dan pembahasan dibawah ini!
a. 4
b. 2
c. -1
d. -4
e. -2
Jawaban: d. -4
Pembahasan :
\(\lim\limits_{x\rightarrow 1}\frac{\mathrm \cos 4x - 1 }{\mathrm x\tan 2x }\)
=\(\lim\limits_{x\rightarrow 1}\frac{-2 \sin^{2}2x}{x \tan 2x}\)
=\(\lim\limits_{x\rightarrow 1}\frac{-2 \sin 2x}{x}\times\frac{\sin 2x }{\tan 2x}\)
=\(-2\times2\times1= -4\)
2. \(\lim\limits_{x\rightarrow 0}\frac{\mathrm x \tan x }{\mathrm 1-\cos 3x }\) =
a. \(\frac{2}{9}\)
b. \(\frac{1}{9}\)
c. \(\frac{4}{9}\)
d. \(\frac{5}{9}\)
e. \(\frac{1}{9}\)
Jawaban: a. \(\frac{2}{9}\)
Pembahasan :
\(\lim\limits_{x\rightarrow 0}\frac{\mathrm x \tan x }{\mathrm 1-\cos 3x }\)
= \(\lim\limits_{x\rightarrow 0}\frac{\mathrm x \tan x }{\mathrm 1 - (1-2 \sin^{2}\frac{3}{2}x)}\)
= \(\lim\limits_{x\rightarrow 0}\frac{\mathrm x \tan x }{\mathrm 2 \sin ^{2}\frac{3}{2} x}\times\frac{x}{x}\)
= \(\frac{1}{2}\times\lim_{x\rightarrow 0}\frac{x}{sin\frac{3}{2}x}\times\frac{x}{sin\frac{3}{2}x}\times\frac{\tan x}{x}\)
= \(\frac{1}{2}\times\lim_{x\rightarrow 0}\frac{x}{sin\frac{3}{2}x}\times\frac{x}{sin\frac{3}{2}x}\times\lim_{x\rightarrow 0}\frac{\tan x}{x}\)
= \(\frac{1}{2}\times\frac{1}{\frac{3}{2}}\times\frac{1}{\frac{3}{2}}\times1\)
= \(\frac{2}{9}\)
3. \(\lim\limits_{x\rightarrow \frac{x}{4}}\frac{\mathrm \cos 2x }{\mathrm \sin x - \cos x }\) =
a. \(3\sqrt{2}\)
b. \(\frac{1}{2}\sqrt{2}\)
c. \(\sqrt\frac{1}{3}\)
d. \(\sqrt{3}\)
e. \(-\sqrt{2}\)
Jawaban: e. \(-\sqrt{2}\)
Pembahasan :
\(\lim\limits_{x\rightarrow \frac{\eta }{4}}\frac{\mathrm \cos 2x }{\mathrm \sin x - \cos x }\)
= \(\lim\limits_{x\rightarrow \frac{\eta }{4}}\frac{\cos^{2}x - \sin^{x}}{\sin x - cos x}\)
= \(\lim\limits_{x\rightarrow \frac{\eta }{4}}\frac{(\cos x - \sin x)(cos x + \sin x)}{-(\cos x - \sin x)} \)
= \(\lim\limits_{x\rightarrow \frac{\eta }{4}}\frac{\cos x+\sin x}{-1}\)
= \(\lim\limits_{x\rightarrow \frac{\eta }{4}}(-\cos x-\sin x) \)
= \(-cos\frac{\eta}{4}-sin\frac{\eta}{4}\)
= \(-\frac{1}{2}\sqrt{2}-\frac{1}{2}\sqrt{2}\)
= \(-\sqrt{2}\)
4. \(\lim\limits_{x\rightarrow 0}\frac{\mathrm 1-\cos 2x }{\mathrm 4x^{2}}\)=
a. \(\frac{1}{3}\)
b. \(\frac{1}{2}\)
c. \(\frac{1}{2}\sqrt{2}\)
d. \(\frac{1}{5}\)
e. \(\frac{1}{2}\sqrt{3}\)
Jawaban: b.\(\frac{1}{2}\)
Pembahasan :
\(\lim_{x\rightarrow 0}\frac{1-\cos 2x}{2x.\sin 2x}\)
= \(\lim_{x\rightarrow 0}\frac{2 \sin^{2} x}{2x.\sin 2x}\)
= \(\lim_{x\rightarrow 0}\frac{2 \sin x}{2x}\times\lim_{x\rightarrow 0}\frac{ \sin x}{2x}\)
= \(\frac{2}{2}\times\frac{1}{2}\)
= \(\frac{1}{2}\)
5. \(\lim\limits_{x\rightarrow 0}\frac{\mathrm x^{2} \sin x + 4 \sin x \sqrt{x}}{\mathrm 4x \tan \sqrt{x} }\) =
a. 4
b. 2
c. 1
d.-4
e.-2
Jawaban: c.1
Pembahasan :
\(\lim\limits_{x\rightarrow 0}\frac{\mathrm x^{2} \sin x + 4 \sin x \sqrt{x}}{\mathrm 4x \tan \sqrt{x}}\)
= \(\lim_{x\rightarrow 0} \frac{x^{2}.x + 4.x\sqrt{x}}{4x.\sqrt{x}}\)
= \(\lim_{x\rightarrow 0} \frac{x^{3} + 4x\sqrt{x}}{4x\sqrt{x}}\)
= \(\lim_{x\rightarrow 0} \frac{x^{3}}{4x\sqrt{x}} + \lim_{x\rightarrow 0}\frac{4x\sqrt{x}}{4x\sqrt{x}}\)
= \(\lim_{x\rightarrow 0}\frac{1}{4}x^{\frac{3}{2}} + \lim_{x\rightarrow 0} 1\)
= 0 + 1
= 1
6. \(\lim\limits_{x\rightarrow 0}\frac{\mathrm -2 \sin^{2} 2x }{\mathrm x\tan 2x }\) =
a.-4
b. 2
c.-1
d. 4
e.-2
Jawaban: a.-4
Pembahasan :
\(\lim\limits_{x\rightarrow 0}\frac{\mathrm -2 \sin^{2} 2x }{\mathrm x\tan 2x }\)
= \(-2\lim_{x\rightarrow 0}\frac{\sin 2x \sin 2x }{x\frac{\sin 2x}{\cos 2x}}\)
= \(-2\lim_{x\rightarrow 0}\frac{\sin 2x \sin 2x }{x}\times\frac{\cos 2x}{\sin 2x}\)
= \(-2\lim_{x\rightarrow 0}\frac{\sin 2x \cos 2x }{x}\times\frac{\sin 2x}{\sin 2x}\)
= \(-2\lim_{x\rightarrow 0}\frac{\sin 2x \cos 2x}{x}\times1\)
= \(-2\lim_{x\rightarrow 0}\frac{\sin 2x}{x}\times\frac{2}{2}\cos 2x\)
= \(-2\times2\lim_{x\rightarrow 0}\frac{\sin 2x}{2x}\times \cos 2x\)
= \(-4 \lim_{x\rightarrow 0}1\times \cos 2x\)
= \(-4 \cos 2\times0\)
= \(-4\times1\)
= -4
7. \(\lim\limits_{x\rightarrow 0}\frac{\mathrm \sin 6x }{\mathrm 2x }\) =
a. 4
b. 2
c. 3
d.-4
e.-2
Jawaban:c.3
Pembahasan :
Misalkan,
6x = U , maka;
x = u. \(\frac{1}{6}\)
= \(\lim\limits_{x\rightarrow 0}\frac{\mathrm \sin 6x }{\mathrm 2x }\)
= \(\lim_{x\rightarrow 0} = \frac{\sin u}{2(\frac{1}{6})}u\)
= \(\lim_{u\rightarrow 0}3\frac{\sin u}{u}\)
= \(3\lim_{u\rightarrow 0}\frac{\sin u}{u}\)
= \(3\times1\)
= 3
8. \(\lim\limits_{x\rightarrow 0}\frac{\mathrm 4x }{\mathrm \sin 3x }\) =
a. \( \frac{4}{3}\)
b. \(\frac{3}{4}\)
c. \(\frac{1}{3}\)
d. \(\frac{1}{4}\)
e. \(\frac{5}{4}\)
Jawaban: a. \(\frac{4}{3}\)
Pembahasan :
\(\lim\limits_{x\rightarrow 0}\frac{\mathrm \tan 4x }{\mathrm \sin 3x }\)
= \(\lim_{x\rightarrow 0}\frac{4}{3}\times\frac{\tan 4x}{4x}\times\frac{3x}{sin3x}\)
= \(\frac{4}{3}\times\lim_{x\rightarrow 0}\frac{\tan 4x}{4x}\times\lim_{x\rightarrow 0}\frac{3x}{\sin 3x}\)
= \(\frac{4}{3}\times1\times1\)
= \(\frac{4}{3}\)
9. \(\lim\limits_{x\rightarrow 0}\frac{\mathrm x \tan 3x }{\mathrm \sin^{2} 6x }\) =
a. \(\frac{1}{3}\)
b. \(\frac{1}{6}\)
c. \(\frac{5}{12}\)
d. \(\frac{1}{4}\)
e. \(\frac{1}{12}\)
Jawaban: e. \(\frac{1}{12}\)
Pembahasan :
\(\lim\limits_{x\rightarrow 0}\frac{\mathrm x \tan 3x }{\mathrm \sin^{2} 6x }\)
= \(\lim_{x\rightarrow 0}\frac{x}{\sin 6x}\times\lim_{x\rightarrow 0}\frac{\tan 3x}{\sin 6x}\)
= \(\frac{1}{6}\times\frac{3}{6}\)
= \(\frac{3}{36}\)
= \(\frac{1}{12}\)
10. \(\lim\limits_{x\rightarrow 0}\frac{\mathrm 5x \tan 8x}{\mathrm 1 - cos 4x}\) =
a. 5
b.-1
c. 0
d. \(\frac{1}{2}\)
e. 4
Jawaban: a. 5
Pembahasan :
\(\lim\limits_{x\rightarrow 0}\frac{\mathrm 5x \tan 8x}{\mathrm 1 - cos 4x}\)
= \(\lim_{x\rightarrow 0}\frac{5x \tan 8x}{1-(1 - 2\sin^{2}2x)}\)
= \(\lim_{x\rightarrow 0}\frac{5x \tan 8x }{2 \sin^{2} 2x}\)
= \(\lim_{x\rightarrow 0} \frac{5x}{2 \sin 2x}\times\frac{\tan 8x}{\sin 2x}\)
= \(\lim_{X\rightarrow 0}\frac{5}{2}\times\frac{2x}{\sin 2x}\times\frac{\tan 8x}{8x}\times\frac{2x}{\sin 2x}\times\frac{8x}{4x}\)
= \(\frac{5}{2}\)
= 5
