sin x / x の極限

2021年9月6日

\displaystyle\lim_{x \rightarrow 0}\ \dfrac{\sin{2x}}{x}

【解答】

\begin{align*} \lim_{x \rightarrow 0}\ \dfrac{\sin{\colorbox{mistyrose}{$2x$}}}{x} &= \lim_{x \rightarrow 0}\ \left(\colorbox{lavender}{$\dfrac{\sin{\colorbox{mistyrose}{$2x$}}}{\colorbox{mistyrose}{$2x$}}$} \times 2\right)\\\\ &= \colorbox{lavender}{$1$} \times2 = 2 \end{align*}

\displaystyle\lim_{x \rightarrow 0}\ \dfrac{\sin{2x}}{\sin{3x}}

【解答】

\begin{align*} & \lim_{x \rightarrow 0}\ \dfrac{\sin{\colorbox{mistyrose}{$2x$}}}{\sin{\colorbox{lightcyan}{$3x$}}}\\\\ &= \lim_{x \rightarrow 0}\ \left(\dfrac{\sin{\colorbox{mistyrose}{$2x$}}}{\sin{\colorbox{lightcyan}{$3x$}}} \times \dfrac{\colorbox{lightcyan}{$3x$}}{\colorbox{mistyrose}{$2x$}} \times \dfrac{2}{3} \right)\\\\ &= \lim_{x \rightarrow 0}\ \left(\colorbox{lightgreen}{$\dfrac{\sin{\colorbox{mistyrose}{$2x$}}}{\colorbox{mistyrose}{$2x$}}$} \times \colorbox{lavender}{$\dfrac{\colorbox{lightcyan}{$3x$}}{\sin{\colorbox{lightcyan}{$3x$}}}$} \times \dfrac23 \right)\\\\ &= \colorbox{lightgreen}{$1$} \times \colorbox{lavender}{$1$} \times \dfrac23 = \dfrac23 \end{align*}

\displaystyle\lim_{x \rightarrow 0}\ \dfrac{\sin{2x}}{3x}

【解答】

\begin{align*} \lim_{x \rightarrow 0}\ \dfrac{\sin{\colorbox{mistyrose}{$2x$}}}{3x} &= \lim_{x \rightarrow 0}\ \left(\dfrac{\sin{\colorbox{mistyrose}{$2x$}}}{\colorbox{mistyrose}{$2x$}} \times \dfrac{2x}{3x}\right)\\\\ &= \lim_{x \rightarrow 0}\ \left(\colorbox{lavender}{$\dfrac{\sin{\colorbox{mistyrose}{$2x$}}}{\colorbox{mistyrose}{$2x$}}$} \times \dfrac{2}{3}\right)\\\\ &= \colorbox{lavender}{$1$} \times \dfrac23 = \dfrac23 \end{align*}

\displaystyle\lim_{x \rightarrow 0}\ \dfrac{\tan{x}}{x}

【解答】

\begin{align*} \lim_{x \rightarrow 0}\ \dfrac{\tan{x}}{x} &= \lim_{x \rightarrow 0}\ \left(\tan{x} \times \dfrac{1}{x}\right)\\\\ &= \lim_{x \rightarrow 0}\ \left(\dfrac{\sin{\colorbox{mistyrose}{$x$}}}{\cos{x}} \times \dfrac{1}{\colorbox{mistyrose}{$x$}}\right)\\\\ &= \lim_{x \rightarrow 0}\ \left(\colorbox{lavender}{$\dfrac{\sin{\colorbox{mistyrose}{$x$}}}{\colorbox{mistyrose}{$x$}}$} \times \dfrac{1}{\cos{x}}\right)\\\\ &= \colorbox{lavender}{$1$} \times \dfrac{1}{\cos{0}} = 1 \times \dfrac{1}{1} = 1 \end{align*}

\displaystyle\lim_{x \rightarrow 0}\ \dfrac{\sin{3x}}{\sin{5x}}

【解答】

\begin{align*} & \lim_{x \rightarrow 0}\ \dfrac{\sin{\colorbox{mistyrose}{$3x$}}}{\sin{\colorbox{lightcyan}{$5x$}}}\\\\ &= \lim_{x \rightarrow 0}\ \left(\dfrac{\sin{\colorbox{mistyrose}{$3x$}}}{\sin{\colorbox{lightcyan}{$5x$}}} \times \dfrac{\colorbox{lightcyan}{$5x$}}{\colorbox{mistyrose}{$3x$}} \times \dfrac{3}{5} \right)\\\\ &= \lim_{x \rightarrow 0}\ \left(\colorbox{lightgreen}{$\dfrac{\sin{\colorbox{mistyrose}{$3x$}}}{\colorbox{mistyrose}{$3x$}}$} \times \colorbox{lavender}{$\dfrac{\colorbox{lightcyan}{$5x$}}{\sin{\colorbox{lightcyan}{$5x$}}}$} \times \dfrac35 \right)\\\\ &= \colorbox{lightgreen}{$1$} \times \colorbox{lavender}{$1$} \times \dfrac35 = \dfrac35 \end{align*}

編集ログ