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データサイエンスのための微分積分 第12回 挟み撃ち定理

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本記事は数学講座2.11挟み撃ち定理を勉強して投稿したメモです。詳細は元の素晴らしい講座のページをチェックしてください。

挟み撃ち定理

  • 関数の挟み撃ち定理
    任意の正の実数$x$に対して、$g(x)\leq f(x) \leq h(x)$且つ$\lim g(x)=\lim h(x)=L$であれば、$\lim f(x)=L$

  • 数列の挟み撃ち定理
    任意な自然数$n$、数列の${a_n},{b_n},{c_n}$、$\exists N\in\mathbb{Z}^+,\forall n > N$の時に、$a_n\le b_n\le c_n$、且つ$\displaystyle\lim_{n\to\infty}a_n=\lim_{n\to\infty}c_n=L$であれば、$\displaystyle\lim_{n\to\infty} b_n=L$
    image.png

sinxがx->0の極限

image.png

cosxがx->0とsinx/xがx->0の極限

  • $\lim_{x\to 0}(1-\cos x)=0\implies \lim_{x\to 0}1-\lim_{x\to 0}\cos x=0\implies\lim_{x\to 0}\cos x=1$
  • $\displaystyle\cos x < \frac{\sin x}{x} < 1$ $\implies $ $\displaystyle\lim_{x\to 0}\frac{\sin x}{x}=1$

練習問題

  • 例1
    image.png

  • 例2
    image.png

  • 例3
    image.png

  • 例4
    image.png

  • 例3と例4:挟み撃ちの両端を探す
    image.png

参照

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