Latihan Matematika Peminatan Kelas XII Mencari Turunan Fungsi Trigonometri
# 3
Pilgan

Nilai dari limxπ3cosxcosπ3tan(xπ3)\lim_{x\to\frac{\pi}{3}}\frac{\cos x-\cos\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)} adalah ....

A

123\frac{1}{2}\sqrt{3}

B

123-\frac{1}{2}\sqrt{3}

C

122\frac{1}{2}\sqrt{2}

D

12\frac{1}{2}

E

12-\frac{1}{2}

Pembahasan:

Akan dicari limxπ3cosxcosπ3tan(xπ3)\lim_{x\to\frac{\pi}{3}}\frac{\cos x-\cos\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}

Misalkan f(x)=cosxf\left(x\right)=\cos x artinya cosπ3=f(π3)\cos\frac{\pi}{3}=f\left(\frac{\pi}{3}\right)

Perlu diingat bahwa untuk sembarang fungsi f(x)f\left(x\right) dan g(x)g\left(x\right) berlaku

limxcf(x)g(x)=limxcf(x)limxcg(x)\lim_{x\to c}f\left(x\right)g\left(x\right)=\lim_{x\to c}f\left(x\right)\lim_{x\to c}g\left(x\right)

Berdasarkan yang diketahui pada soal dan pemisalan yang dibuat, diperoleh

limxπ3cosxcosπ3tan(xπ3)=limxπ3f(x)f(π3)tan(xπ3)\lim_{x\to\frac{\pi}{3}}\frac{\cos x-\cos\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}=\lim_{x\to\frac{\pi}{3}}\frac{f\left(x\right)-f\left(\frac{\pi}{3}\right)}{\tan\left(x-\frac{\pi}{3}\right)}

limxπ3cosxcosπ3tan(xπ3)=limxπ3xπ3tan(xπ3).f(x)f(π3)xπ3\Leftrightarrow\lim_{x\to\frac{\pi}{3}}\frac{\cos x-\cos\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}=\lim_{x\to\frac{\pi}{3}}\frac{x-\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}.\frac{f\left(x\right)-f\left(\frac{\pi}{3}\right)}{x-\frac{\pi}{3}}

limxπ3cosxcosπ3tan(xπ3)=limxπ3xπ3tan(xπ3).limxπ3f(x)f(π3)xπ3\Leftrightarrow\lim_{x\to\frac{\pi}{3}}\frac{\cos x-\cos\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}=\lim_{x\to\frac{\pi}{3}}\frac{x-\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}.\lim_{x\to\frac{\pi}{3}}\frac{f\left(x\right)-f\left(\frac{\pi}{3}\right)}{x-\frac{\pi}{3}}

Perlu diingat pula bahwa limxcxatan(xa)=1\lim_{x\to c}\frac{x-a}{\tan\left(x-a\right)}=1 dan limxcf(x)f(c)xc=f(c)\lim_{x\to c}\frac{f\left(x\right)-f\left(c\right)}{x-c}=f'\left(c\right)

sehingga didapat

limxπ3cosxcosπ3tan(xπ3)=1.f(π3)\lim_{x\to\frac{\pi}{3}}\frac{\cos x-\cos\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}=1.f'\left(\frac{\pi}{3}\right)

Karena f(x)=cosxf\left(x\right)=\cos x maka f(x)=sinxf'\left(x\right)=-\sin x. Akibatnya

f(π3)=sinπ3=123f'\left(\frac{\pi}{3}\right)=-\sin\frac{\pi}{3}=-\frac{1}{2}\sqrt{3}