ネイピア数と極限

ネイピア数と極限

(1)

\[ \lim_{h\rightarrow0}\left(1-h\right)^{\frac{1}{h}}=\frac{1}{e} \]

(2)

\[ \lim_{h\rightarrow0}\left(1+ah\right)^{\frac{1}{h}}=e^{a} \]

(3)

\[ \lim_{h\rightarrow0}\frac{e^{h}-1}{h}=1 \]

(4)

\[ \lim_{h\rightarrow0}\frac{\log\left(1+h\right)}{h}=1 \]

(1)

\begin{align*} \lim_{h\rightarrow0}\left(1-h\right)^{\frac{1}{h}} & =\lim_{h\rightarrow0}\left(\frac{1}{1-h}\right)^{-\frac{1}{h}}\\ & =\lim_{h\rightarrow0}\left(1+\frac{h}{1-h}\right)^{-\frac{1}{h}}\\ & =\lim_{h\rightarrow0}\frac{1}{\left(1+h\right)^{\frac{1}{h}}}\\ & =\frac{1}{e} \end{align*}

(2)

\begin{align*} \lim_{h\rightarrow0}\left(1+ah\right)^{\frac{1}{h}} & =\lim_{h\rightarrow0}\left(1+ah\right)^{\frac{a}{ah}}\\ & =e^{a} \end{align*}

(3)

\begin{align*} \lim_{h\rightarrow0}\frac{e^{h}-1}{h} & =\lim_{h\rightarrow0}\frac{\left(e^{h}-1\right)^{\prime}}{\left(h\right)^{\prime}}\\ & =\lim_{h\rightarrow0}e^{h}\\ & =1 \end{align*}

(3)-2

\begin{align*} \lim_{h\rightarrow0}\frac{e^{h}-1}{h} & =\lim_{h\rightarrow0}\frac{\left(1+h\right)^{\frac{h}{h}}-1}{h}\\ & =\lim_{h\rightarrow0}1\\ & =1 \end{align*}

(4)

\begin{align*} \lim_{h\rightarrow0}\frac{\log\left(1+h\right)}{h} & =\lim_{h\rightarrow0}\frac{\left(\log\left(1+h\right)\right)^{\prime}}{\left(h\right)^{\prime}}\\ & =\lim_{h\rightarrow0}\frac{\left(1+h\right)^{-1}}{1}\\ & =\lim_{h\rightarrow0}\frac{1}{1+h}\\ & =1 \end{align*}

(4)-2

\begin{align*} \lim_{h\rightarrow0}\frac{\log\left(1+h\right)}{h} & =\lim_{h\rightarrow0}\log\left(1+h\right)^{\frac{1}{h}}\\ & =\log\left(\lim_{h\rightarrow0}\left(1+h\right)^{\frac{1}{h}}\right)\\ & =\log e\\ & =1 \end{align*}

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ネイピア数と極限

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