sin x / x の極限

【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*}

【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*}

【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*}

【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*}

【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*}
  • 20210905…初版公開。問題数5。時間が出来たら公式の証明などを入れる>私

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