不定形の極限値(limit of indeterminate forms)

確認問題

1.

次の極限値を求めよう．

(a) $\displaystyle{\lim_{x \rightarrow 0+}\frac{\sin{x}}{\sqrt{x}}}$ (b) $\displaystyle{\lim_{x \rightarrow 1}\frac{\log{x}}{1-x}}$ (c) $\displaystyle{\lim_{x \rightarrow 4}\frac{\sqrt{x}-2}{x-4}}$ (d) $\displaystyle{\lim_{x \rightarrow 0}\frac{2^{x} - 1}{x}}$

(e) $\displaystyle{\lim_{x \rightarrow 0}\frac{1 - \cos{x}}{3x}}$ (f) $\displaystyle{\lim_{x \rightarrow \infty}\frac{x-1}{x+1}}$ (g) $\displaystyle{\lim_{x \rightarrow \infty}\frac{2\sin{x}}{x}}$ (h) $\displaystyle{\lim_{x \rightarrow 0}\frac{e^{x} - e^{-x}}{x}}$

演習問題

1.

次の極限値を求めよう．

(a) $\displaystyle{\lim_{x \rightarrow 0}\frac{\sin{2x}}{\sin{3x}}}$ (b) $\displaystyle{\lim_{x \rightarrow 0}\frac{\cos{x} - 1}{x}}$ (c) $\displaystyle{\lim_{x \rightarrow 0}\frac{\sin^{-1}{x}}{x}}$ (d) $\displaystyle{\lim_{x \rightarrow 0}x\sin{\frac{1}{x}}}$

(e) $\displaystyle{\lim_{x \rightarrow 0}(\frac{1}{x^{2}} - \frac{1}{\sin^{2}{x}})}$ (f) $\displaystyle{\lim_{x \rightarrow \frac{\pi}{2}}(1 - \sin{x})^{\cos{x}}}$ (g) $\displaystyle{\lim_{x \rightarrow 0}(\frac{e^{x} - 1}{x})^{1/x}}$

(h) $\displaystyle{\lim_{x \rightarrow 0}(1 - x)^{\frac{1}{\sin{x}}}}$