Diberikan k (θ) = cos² 2θMaka,.[tex] \displaystyle \rm\lim_{x \to 0}

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Diberikan k (θ) = cos² 2θMaka,
.
 \displaystyle \rm\lim_{x \to 0} \frac{k( \frac{ π}{6} + x)- k( \frac{π}{6} ) }{x} =
.
A). 1/2√3
B). -√3
C). 1/2
D). 1
A). 0​

Jawaban dan Penjelasan

Berikut ini adalah pilihan jawaban terbaik dari pertanyaan diatas.

Nilai dari \displaystyle{ \lim_{x \to 0} \frac{k\left ( \frac{\pi}{6}+x \right )-k\left ( \frac{\pi}{6} \right )}{x} }adalahB). -√3.

PEMBAHASAN

Teorema pada limit adalah sebagai berikut :

(i)~\lim\limits_{x \to c} f(x)=f(c)

(ii)~\lim\limits_{x \to c} kf(x)=k\lim\limits_{x \to c} f(x)

(iii)~\lim\limits_{x \to c} [f(x)\pm g(x)]=\lim\limits_{x \to c} f(x)\pm\lim\limits_{x \to c} g(x)

(iv)~\lim\limits_{x \to c} [f(x)\times g(x)]=\lim\limits_{x \to c} f(x)\times\lim\limits_{x \to c} g(x)

(v)~\lim\limits_{x \to c} \left [ \frac{f(x)}{g(x)} \right ]=\frac{\lim\limits_{x \to c} f(x)}{\lim\limits_{x \to c} g(x)}

(vi)~\lim\limits_{x \to c} \left [ f(x) \right ]^n=\left [ \lim\limits_{x \to c} f(x) \right ]^n

Rumus untuk limit fungsi trigonometri :

(i)~\lim\limits_{x \to 0} \frac{sinax}{bx}=\lim\limits_{x \to 0} \frac{tanax}{bx}=\frac{a}{b}

(ii)~\lim\limits_{x \to 0} \frac{ax}{sinbx}=\lim\limits_{x \to 0} \frac{ax}{tanbx}=\frac{a}{b}

(iii)~\lim\limits_{x \to 0} \frac{sinax}{sinbx}=\lim\limits_{x \to 0} \frac{tanax}{tanbx}=\frac{a}{b}

(iv)~\lim\limits_{x \to a} \frac{sin(x-a)}{(x-a)}=\lim\limits_{x \to a} \frac{tan(x-a)}{(x-a)}=1

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DIKETAHUI

k(\theta)=cos^2(2\theta)

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DITANYA

Tentukan nilai dari \displaystyle{ \lim_{x \to 0} \frac{k\left ( \frac{\pi}{6}+x \right )-k\left ( \frac{\pi}{6} \right )}{x} }.

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PENYELESAIAN

\displaystyle{ \lim_{x \to 0} \frac{k\left ( \frac{\pi}{6}+x \right )-k\left ( \frac{\pi}{6} \right )}{x} }

\displaystyle{=\lim_{x \to 0} \frac{cos^2\left [ 2\left ( \frac{\pi}{6}+x \right ) \right ]-cos^2\left [ 2\left ( \frac{\pi}{6} \right ) \right ]}{x} }

\displaystyle{=\lim_{x \to 0} \frac{cos^2\left ( \frac{\pi}{3}+2x \right )-cos^2\left ( \frac{\pi}{3} \right )}{x} }

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.~~~~~~~~~~\boldsymbol{a^2-b^2=(a+b)(a-b)}

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\displaystyle{=\lim_{x \to 0} \frac{\left [ cos\left ( \frac{\pi}{3}+2x \right )+cos\left ( \frac{\pi}{3} \right ) \right ]\left [ cos\left ( \frac{\pi}{3}+2x \right )-cos\left ( \frac{\pi}{3} \right ) \right ]}{x} }

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.~~~~~~~~~~\boldsymbol{cos\alpha+cos\beta=2cos\left ( \frac{\alpha+\beta}{2} \right )cos\left ( \frac{\alpha-\beta}{2} \right )}

.~~~~~~~~~~\boldsymbol{cos\alpha-cos\beta=-2sin\left ( \frac{\alpha+\beta}{2} \right )sin\left ( \frac{\alpha-\beta}{2} \right )}

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\displaystyle{=\lim_{x \to 0} \frac{\left [ 2cos\left ( \frac{\frac{\pi}{3}+2x+\frac{\pi}{3}}{2} \right )cos\left ( \frac{\frac{\pi}{3}+2x-\frac{\pi}{3}}{2} \right ) \right]\left [ -2sin\left ( \frac{\frac{\pi}{3}+2x+\frac{\pi}{3}}{2} \right )sin\left ( \frac{\frac{\pi}{3}+2x-\frac{\pi}{3}}{2} \right ) \right]}{x} }

\displaystyle{=\lim_{x \to 0} \frac{-4\left [ cos\left ( \frac{\pi}{3}+x \right )cos\left ( x \right ) \right ]\left [ sin\left ( \frac{\pi}{3}+x \right )sin\left ( x \right ) \right ]}{x} }

\displaystyle{=-4\lim_{x \to 0} cos\left ( \frac{\pi}{3}+x \right )cos\left ( x \right )sin\left ( \frac{\pi}{3}+x \right )\times\lim_{x \to 0}\frac{sin(x)}{x} }

\displaystyle{=-4\times cos\left ( \frac{\pi}{3}+0 \right )cos\left ( 0 \right )sin\left ( \frac{\pi}{3}+0 \right )\times1 }

\displaystyle{=-4\times cos\left ( \frac{\pi}{3}\right )\times1\times sin\left ( \frac{\pi}{3} \right ) }

\displaystyle{=-4\times\left ( \frac{1}{2}\right )\times\left ( \frac{\sqrt{3}}{2} \right ) }

\displaystyle{=-\sqrt{3} }

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KESIMPULAN

Nilai dari \displaystyle{ \lim_{x \to 0} \frac{k\left ( \frac{\pi}{6}+x \right )-k\left ( \frac{\pi}{6} \right )}{x} }adalahB). -√3.

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PELAJARI LEBIH LANJUT

  1. Limit fungsi trigonometri : yomemimo.com/tugas/41998117
  2. Limit trigonometri : yomemimo.com/tugas/38915095
  3. Limit trigonometri : yomemimo.com/tugas/30308496

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DETAIL JAWABAN

Kelas : 11

Mapel: Matematika

Bab : Limit Fungsi

Kode Kategorisasi: 11.2.8

Kata Kunci : limit, fungsi, trigonometri.

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Last Update: Sun, 30 Jan 22