関数の極限②指数・対数

\begin{align*}
\lim_{x \rightarrow -\infty}\ 2^x
&\textcolor{orange}{= 2^{-\infty}}\\
&= 0
\end{align*}

\begin{align*}
\lim_{x \rightarrow \infty}\ \left(\dfrac13\right)^x
&\textcolor{orange}{= \left(3^{-1}\right)^{\infty}} = 3^{-\infty}\\
&= 0
\end{align*}

\begin{align*}
\lim_{x \rightarrow \infty}\ \log_{2}x
&\textcolor{orange}{= \log_{2}\infty}\\
&= \infty
\end{align*}

\begin{align*}
\lim_{x \rightarrow +0}\ \log_{0.5}x
&\textcolor{orange}{= \log_{0.5}(+0)}\\
&= \infty
\end{align*}

\begin{align*}
\lim_{x \rightarrow \infty}\ 2^{-2x}
&\textcolor{orange}{= 2^{-2\infty} = 2^{-\infty}}\\
&= 0
\end{align*}

\lim_{x \rightarrow \infty}\ \log_{2}\dfrac{2x+3}{x}
= \log_{2}\dfrac{2\infty+3}{\infty} = \log_{2}\colorbox{red}{$\dfrac{\infty}{\infty}$}

\begin{align*}
\lim_{x \rightarrow \infty}\ \log_{2}\dfrac{2x+3}{x}
&\textcolor{orange}{= \lim_{x \rightarrow \infty}\ \log_{2}\dfrac{\ \dfrac{2x}{x}+\dfrac{3}{x}\ }{\dfrac{x}{x}}}\\\\
&= \lim_{x \rightarrow \infty}\ \log_{2}\dfrac{2+\dfrac{3}{x}}{1}\\\\
&= \lim_{x \rightarrow \infty}\ \log_{2}\ \left(2+\dfrac{3}{x}\right)\\\\
&\textcolor{orange}{= \log_{2}\left(2+\colorbox{lightcyan}{$\dfrac{3}{\infty}$}\right)}\\\\
&\textcolor{orange}{=\log_{2}(2+\colorbox{lightcyan}{$0$})}\\\\
&= \log_{2}2 = 1
\end{align*}

\begin{align*}
\lim_{x \rightarrow \infty}\ 2^{-3x}
&\textcolor{orange}{= 2^{-3\infty} = 2^{-\infty}}\\
&= 0
\end{align*}

\begin{align*}
\lim_{x \rightarrow -\infty}\ 2^{-2x}
&\textcolor{orange}{= 2^{-2(-\infty)} = 2^{\infty}}\\
&= \infty
\end{align*}

\lim_{x \rightarrow \infty}\ \log_{2}\dfrac{4x-1}{x+2}
= \log_{2}\dfrac{4\infty-1}{\infty+2} = \log_{2}\colorbox{red}{$\dfrac{\infty}{\infty}$}

\begin{align*}
\lim_{x \rightarrow \infty}\ \log_{2}\dfrac{4x-1}{x+2}
&\textcolor{orange}{= \lim_{x \rightarrow \infty}\ \log_{2}\dfrac{\ \dfrac{4x}{x}-\dfrac{1}{x}\ }{\dfrac{x}{x}+\dfrac{2}{x}}}\\\\
&= \lim_{x \rightarrow \infty}\ \log_{2}\dfrac{4-\dfrac{1}{x}}{1+\dfrac{2}{x}}\\\\
&\textcolor{orange}{= \log_{2}\dfrac{4-\colorbox{lightcyan}{$\dfrac{1}{\infty}$}}{1+\colorbox{lightcyan}{$\dfrac{2}{\infty}$}}}\\\\
&\textcolor{orange}{=\log_{2}\dfrac{4-\colorbox{lightcyan}{$0$}}{1+\colorbox{lightcyan}{$0$}}}\\\\
&= \log_{2}4\\\\
&= \log_{2}{2^2} = 2
\end{align*}

\lim_{x \rightarrow 0}\ \sin{x} =\sin{0}=0

\lim_{x \rightarrow 0}\ \cos{x} =\cos{0}=1

\lim_{x \rightarrow \infty}\ \sin\dfrac{1}{x} \color{orange}=\sin\dfrac{1}{\infty}\color{black}=\sin{0}=0

\lim_{x \rightarrow \infty}\ \cos\dfrac{1}{x} \color{orange}=\cos\dfrac{1}{\infty}\color{black}=\cos{0}=1

\lim_{x \rightarrow -\infty}\ \sin\dfrac{1}{x} \color{orange}=\sin\dfrac{1}{-\infty}\color{black}=\sin{0}=0

\lim_{x \rightarrow -\infty}\ \cos\dfrac{1}{x} \color{orange}=\cos\dfrac{1}{-\infty}\color{black}=\cos{0}=1

\lim_{x \rightarrow \pi}\ \tan{x} \color{black}=\tan{\pi}=0