# 不定型極限問題 (Indeterminate Form)

Find (a) $$\displaystyle \lim_{x \to 0} \frac{x^2}{x}$$ (b) $$\displaystyle \lim_{x \to 0} \frac{x}{x^2}$$ (c) $$\displaystyle \lim_{x \to 0^+} \frac{x}{x^2}$$

# L’Hôpital’s Rule

L’Hôpital’s Rule的定理敘述如下。

Suppose $$f$$ and $$g$$ are differentiable and $$g'(x) \neq 0$$ on an open interval $$I$$ that contains $$a$$ (except possibly at $$a$$). Suppose that $\lim_{x \to a} f(x) = 0 \quad \text{and} \quad \lim_{x \to a} g(x) = 0$ or that $\lim_{x \to a} f(x) = \pm \infty \quad \text{and} \quad \lim_{x \to a} g(x) = \pm \infty$ Then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ if the limit on the right side exists (or is $$\pm \infty$$).