1. [tex]\lim_{x \to \0} 0 \frac{cos 4x-cos 2x}{ x^{2} } [/tex] 2. [tex] \lim_{x \to \0} 0 \frac{sin 2x-tan 2x}{ x^{3} } [/tex] 3. [tex] \lim_{x \to \0} 0 \
Matematika
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Pertanyaan
1. [tex]\lim_{x \to \0} 0 \frac{cos 4x-cos 2x}{ x^{2} } [/tex]
2. [tex] \lim_{x \to \0} 0 \frac{sin 2x-tan 2x}{ x^{3} } [/tex]
3. [tex] \lim_{x \to \0} 0 \frac{1- \sqrt{cos x} }{ x^{2} } [/tex]
2. [tex] \lim_{x \to \0} 0 \frac{sin 2x-tan 2x}{ x^{3} } [/tex]
3. [tex] \lim_{x \to \0} 0 \frac{1- \sqrt{cos x} }{ x^{2} } [/tex]
1 Jawaban
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1. Jawaban whongaliem
[tex]1) \lim_{x \to \ 0} \frac{cos 4x - cos 2x}{ x^{2} } = \lim_{x \to \ 0} \frac{- 2sin 3x . sin 2x }{ x^{2} } [/tex]
[tex]= - 2 \lim_{x \to \ 0} \frac{sin 3x}{x} \frac{sin 2x}{x} [/tex]
= - 2 . 3 . 2
= - 12
[tex]2) \lim_{x \to \ 0} \frac{sin 2x - tan 2x}{ x^{3} } = \lim_{x \to \ 0} \frac{sin 2x - \frac{sin 2x}{cos 2x} }{ x^{3} } [/tex]
[tex]= \lim_{x \to \ 0} \frac{sin 2x . cos 2x - sin 2x}{ x^{3} cox 2x} [/tex]
[tex]= \lim_{x \to \ 0} \frac{sin 2x (cos 2x - 1)}{ x^{3} cos 2x} [/tex]
[tex]= \lim_{x \to \ 0} \frac{sin 2x ( cos^{2} x - sin^{2} x - 1) }{ x^{3} cos 2x} [/tex]
[tex] \lim_{x \to \ 0} \frac{sin 2x (1 - sin^{2} x - sin^{2} x - 1)}{ x^{3} cos 2x} [/tex]
[tex]= \lim_{x \to \ 0} \frac{sin 2x . - 2 sin^{2} x}{ x^{3} cos 2x} [/tex]
[tex]= - 2 \lim_{x \to \ 0} \frac{sin 2x}{x} \frac{ sin^{2} x }{ x^{2} } \frac{1}{cos 2x} [/tex]
= - 2 . 2 . 1² . 1
= - 4
[tex]3) \lim_{x \to \ 0} \frac{1 - \sqrt{cos x} }{ x^{2} } X \frac{1 + \sqrt{cos x} }{1 + \sqrt{cos x} } = \lim_{x \to \ 0} \frac{1 - cos x}{ x^{2} (1 + \sqrt{cos x}) } [/tex]
[tex]= \lim_{x \to \ 0} \frac{1 - cos^{2} \frac{1}{2} x + sin^{2} \frac{1}{2} x}{ x^{2} (1 + \sqrt{cos x} )} [/tex]
[tex] \lim_{x \to \ 0} \frac{ sin^{2} \frac{1}{2} x + sin^{2} \frac{1}{2}x }{ x^{2} (1 + \sqrt{cos x} } [/tex]
[tex]= \lim_{x \to \ 0} \frac{2. sin^{2} \frac{1}{2} x}{ x^{2} (1 + \sqrt{cos x} )} [/tex]
[tex]= 2 . \lim_{x \to \0} \frac{ sin^{2} \frac{1}{2} x}{ x^{2} } } \frac{1}{1 + \sqrt{cos x} } [/tex]
[tex]= 2 . (\frac{1}{2}) ^{2} . \frac{1}{ 1 + 1} [/tex]
[tex]= 2 . \frac{1}{4} . \frac{1}{2} [/tex]
[tex]= \frac{1}{4} [/tex]