[tex]\begin{aligned}&\textsf{Nilai dari }\lim_{x\to0}\frac{1-\cos^2x}{x^2\tan\left(x+\dfrac{\pi}{3}\right)}\\&\sf adalah\ \:\boxed{\,\bf\frac{1}{3}\sqrt{3}\,}\quad(\sf opsi\ c.)\end{aligned}[/tex]
Limit Fungsi Trigonometri
[tex]\begin{aligned}&\lim_{x\to0}\frac{1-\cos^2x}{x^2\tan\left(x+\dfrac{\pi}{3}\right)}\\\vphantom{\bigg|}&\quad...\ \left[\ 1-\cos^2x=\sin^2x\ \right]\\&{=\ }\lim_{x\to0}\frac{\sin^2x}{x^2\tan\left(x+\dfrac{\pi}{3}\right)}\\&{=\ }\lim_{x\to0}\left[\frac{\sin^2x}{x^2}\cdot\frac{1}{\tan\left(x+\dfrac{\pi}{3}\right)}\right]\\\vphantom{\Big|}&\quad...\ \textsf{Dengan pengecualian pada bentuk tak tentu:}\end{aligned}[/tex][tex]\begin{aligned}&{=\ }\lim_{x\to0}\frac{\sin^2x}{x^2}\cdot\lim_{x\to0}\frac{1}{\tan\left(x+\dfrac{\pi}{3}\right)}\\&{=\ }\lim_{x\to0}\left(\frac{\sin x}{x}\right)^2\cdot\frac{1}{\tan\left(\dfrac{\pi}{3}\right)}\\\vphantom{\Bigg|}&\quad...\ \left[\ \tan\left(\dfrac{\pi}{3}\right)=\sqrt{3}\ \right]\\&{=\ }\left(\lim_{x\to0}\frac{\sin x}{x}\right)^2\cdot\frac{1}{\sqrt{3}}\\\vphantom{\Bigg|}&\quad...\ \left[\ \lim_{x\to0}\frac{\sin x}{x}=1\ \right]\end{aligned}[/tex][tex]\begin{aligned}&{=\ }1^2\cdot\frac{1}{\sqrt{3}}\:=\:\frac{1}{\sqrt{3}}\\&{=\ }\frac{1}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}\\&{=\ }\boxed{\,\bf\frac{1}{3}\sqrt{3}\,}\end{aligned}[/tex]
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[tex]\begin{aligned}&\textsf{Nilai dari }\lim_{x\to0}\frac{1-\cos^2x}{x^2\tan\left(x+\dfrac{\pi}{3}\right)}\\&\sf adalah\ \:\boxed{\,\bf\frac{1}{3}\sqrt{3}\,}\quad(\sf opsi\ c.)\end{aligned}[/tex]
Penjelasan dengan langkah-langkah:
Limit Fungsi Trigonometri
[tex]\begin{aligned}&\lim_{x\to0}\frac{1-\cos^2x}{x^2\tan\left(x+\dfrac{\pi}{3}\right)}\\\vphantom{\bigg|}&\quad...\ \left[\ 1-\cos^2x=\sin^2x\ \right]\\&{=\ }\lim_{x\to0}\frac{\sin^2x}{x^2\tan\left(x+\dfrac{\pi}{3}\right)}\\&{=\ }\lim_{x\to0}\left[\frac{\sin^2x}{x^2}\cdot\frac{1}{\tan\left(x+\dfrac{\pi}{3}\right)}\right]\\\vphantom{\Big|}&\quad...\ \textsf{Dengan pengecualian pada bentuk tak tentu:}\end{aligned}[/tex]
[tex]\begin{aligned}&{=\ }\lim_{x\to0}\frac{\sin^2x}{x^2}\cdot\lim_{x\to0}\frac{1}{\tan\left(x+\dfrac{\pi}{3}\right)}\\&{=\ }\lim_{x\to0}\left(\frac{\sin x}{x}\right)^2\cdot\frac{1}{\tan\left(\dfrac{\pi}{3}\right)}\\\vphantom{\Bigg|}&\quad...\ \left[\ \tan\left(\dfrac{\pi}{3}\right)=\sqrt{3}\ \right]\\&{=\ }\left(\lim_{x\to0}\frac{\sin x}{x}\right)^2\cdot\frac{1}{\sqrt{3}}\\\vphantom{\Bigg|}&\quad...\ \left[\ \lim_{x\to0}\frac{\sin x}{x}=1\ \right]\end{aligned}[/tex]
[tex]\begin{aligned}&{=\ }1^2\cdot\frac{1}{\sqrt{3}}\:=\:\frac{1}{\sqrt{3}}\\&{=\ }\frac{1}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}\\&{=\ }\boxed{\,\bf\frac{1}{3}\sqrt{3}\,}\end{aligned}[/tex]