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*}ページ情報
| タイトル | ネイピア数と極限 |
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対数となる極限
\[
\lim_{\alpha\rightarrow-1}\frac{z^{\alpha+1}}{\alpha+1}=\Log\left(z\right)+\lim_{\alpha\rightarrow-1}\frac{1}{\alpha+1}
\]
0の0乗の極限
\[
\lim_{x\rightarrow0}x^{\left|a\right|}=\delta_{0a}
\]