contoh soal yang kayak gini tuh gimana ya?tolong bantu jawab ya
Jawab:
[tex]\large\text{$\begin{aligned}&\lim_{x\to\,0}\frac{\sin^2(ax)}{(bx)^2}=\bf\frac{a^2}{b^2}\end{aligned}$}[/tex]
Penjelasan dengan langkah-langkah:
Limit Fungsi Trigonometri
[tex]\large\text{$\begin{aligned}&{\lim_{x\to\,0}\frac{\sin^2(ax)}{(bx)^2}}\\&{=\ }\lim_{x\to\,0}\frac{\sin^2(ax)}{b^2x^2}\\&{=\ }\lim_{x\to\,0}\left(\frac{1}{b^2}\cdot\frac{\sin^2(ax)}{x^2}\right)\\&{\quad}\left[\ \normalsize\text{$\begin{aligned}&\textsf{$\frac{1}{b^2}$ adalah konstanta.}\\&\textsf{Jadikan konstanta pengali limit.}\end{aligned}$}\right.\\&{=\ }\frac{1}{b^2}\cdot\lim_{x\to\,0}\frac{\sin^2(ax)}{x^2}\end{aligned}$}[/tex]
[tex]\large\text{$\begin{aligned}&{=\ }\frac{1}{b^2}\cdot\lim_{x\to\,0}\left(\frac{\sin(ax)}{x}\right)^2\\&{\quad}\left[\ \normalsize\text{$\begin{aligned}&\textsf{Karena ini bentuk tak tentu,}\\&\textsf{maka berlaku:}\\&\lim_{x\to\,a}\left[f(x)\right]^b=\left[\lim_{x\to\,a}f(x)\right]^b\end{aligned}$}\right.\\&{=\ }\frac{1}{b^2}\left(\lim_{x\to\,0}\frac{\sin(ax)}{x}\right)^2\end{aligned}$}[/tex]
[tex]\large\text{$\begin{aligned}&{\quad}\left[\ \normalsize\text{$\begin{aligned}&\lim_{x\to\,0}\sin(ax)=\sin0=0\\&\lim_{x\to\,0}x=0\\&\textsf{Karena $\frac{0}{0}$ masih merupakan}\\&\textsf{bentuk tak tentu, maka terapkan}\\&\textsf{dalil L'H\^opital.}\end{aligned}$}\right.\\&{=\ }\frac{1}{b^2}\left(\lim_{x\to\,0}\frac{\frac{d}{dx}\sin(ax)}{\frac{d}{dx}x}\right)^2\\&{=\ }\frac{1}{b^2}\left(\lim_{x\to\,0}\frac{\cos(ax)\left(\frac{d}{dx}ax\right)}{1}\right)^2\end{aligned}$}[/tex]
[tex]\large\text{$\begin{aligned}&{=\ }\frac{1}{b^2}\left(\lim_{x\to\,0}\frac{a\cos(ax)}{1}\right)^2\\&{=\ }\frac{1}{b^2}\left(\lim_{x\to\,0}a^2\cos^2(ax)\right)\\&{\quad}\left[\ \normalsize\text{$\begin{aligned}&\textsf{$a^2$ adalah konstanta.}\\&\textsf{Jadikan konstanta pengali limit.}\end{aligned}$}\right.\\&{=\ }\frac{a^2}{b^2}\left(\lim_{x\to\,0}\cos^2(ax)\right)\\&{\quad}\left[\ \normalsize\text{$\begin{aligned}&\textsf{Substitusi $x$ dengan $0$}\\\end{aligned}$}\right.\end{aligned}$}[/tex]
[tex]\large\text{$\begin{aligned}&{=\ }\frac{a^2}{b^2}\cdot\cos^2(a\cdot0)\\&{=\ }\frac{a^2}{b^2}\cdot\cos^2(0)\\&{=\ }\frac{a^2}{b^2}\cdot1\\&{=\ }\frac{a^2}{b^2}\\\\&{\therefore\quad}\boxed{\ \lim_{x\to\,0}\frac{\sin^2(ax)}{(bx)^2}=\bf\frac{a^2}{b^2}\ }\end{aligned}$}[/tex]
Catatan:
Pada penyelesaian di atas, saya coba merinci langkah-langkahnya. Kalau sudah terbiasa, tidak akan sepanjang itu.
