L'Hopital's Rule

Cauchy 均值定理的一個應用, 是 l'Hopital 求極限的法則.

定理. 設 f(x) 及 g(x) 在包含 c 點的一間隔中連續且可微分. 假定 g' 在此間隔中 $\neq{0},f(c)=g(c)=0$, 且

$$\lim_{x \rightarrow c}\frac{f'(x)}{g'(x)}=\ell,$$

則

$$\lim_{x \rightarrow c}\frac{f(x)}{g(x)}=\ell.$$

證明. From Cauchy's Theorem,

$$\frac{f(x)}{g(x)}=\frac{f(x)-f(c)}{g(x)-g(c)}=\frac{f'(\xi)}{g'(\xi)}$$

for some number $\xi$ between x and c. As $x\rightarrow{c}$, $\xi$ is squeezed to c too. Hence the theorem.

注意. 在本定理的證明中, 其實我們沒用到 f 和 g 在 c 點的可微分性, 所以定理的條件有一點多餘.

One or both of the numbers c and l can be changed to $\pm\infty$, but the grounds of these cases lie much deeper, and for the time being we just take them for granted and leave their proofs till a later day.

This theorem, as well as the cases with c and/or l changed to $\pm\infty$, is known as l'Hopital's rule. We give some applications below:

例 1   Compute $$\lim_{x \rightarrow 0}\frac{1-\cos x}{x^{2}}$$.

Solution. Since $1-\cos 0=0$, we may apply l'Hopital's rule to get

$$\displaystyle\lim_{x \rightarrow 0}\frac{1-\cos x}{x^{2}} =\lim_{x \rightarrow 0}\frac{\sin x}{2x}=\frac{1}{2}.$$

例 2   Compute $$\lim_{x \rightarrow \infty}{(x^{2}+1)}^{1/\ln{x}}$$.

Solution. Let $f(x)=(x^{2}+1)^{1/\ln{x}}$. Then

$$\ln{f(x)}=\frac{\ln{(x^{2}+1)}}{\ln{x}}.$$

By l'Hopital's rule we get

$$\lim_{x \rightarrow \infty}{\ln{f(x)}}= \lim_{x \rightarrow \infty}\frac{2x/(x^{2}+1)}{1/x}=2.$$

Therefore

$$\lim_{x \rightarrow \infty}{f(x)}=e^{2}.$$

當 $x\rightarrow{c}$ 求 f(x) 的極限時, 如果我們把 f(x) 的定義域局限在 x 的左邊, 則所求的極限叫 f 的左翼極限, 以符號 $\lim_{x \rightarrow c^{-}}{f(x)}$ 表之. 仿此如果我們把 f(x) 的定義域局限在 x 的右邊, 則得到 f 的右翼極限 $\lim_{x \rightarrow c^{+}}{f(x)}$. 左翼和右翼極限通稱單翼極限, 對於它們, l'Hopital's rule 也管用.

例 3   設 $\theta >0$.試證 (a) $\lim_{x \rightarrow 0^{+}}{x^{\theta}}{\ln{x}}=0$, (b) $\lim_{x \rightarrow \infty} \displaystyle\frac{\ln{x}}{{x}^{\theta}}=0$.

證明. (a) $$\lim_{x \rightarrow 0^{+}}{x^{\theta}}{\ln{x}}= \lim_{x \rightarro... ...eta}-1}}= \lim_{x \rightarrow 0^{+}}\displaystyle\frac{x^{\theta}}{-{\theta}}=0$$,
(b) $$\lim_{x \rightarrow \infty}\displaystyle\frac{\ln{x}}{x^{\theta}}=... ...1}}= \lim_{x \rightarrow \infty}\displaystyle\frac{1}{{\theta}{x^{\theta}}}=0.$$

習$\quad$題

1.

參考本節中討論微分學的均值定理前的文字,利用微分學均值定理證明 積分學均值定理,並注意 $\xi$ 可取自開間隔 (a,b).

2.

對下列各函數, 在其定義域中找出滿足 mean-value theorem 的 $\xi$.
(a) f(x)=x² on [1,2],     (b) f(x)=1/x on [4,9],
(c) f(x)=x on [0,1],     (d) f(x)=2x³-x²+3x-5 on [-1,1].

3.

設 a, b 是任意固定的實數, 試證下二不等式: (a) $\vert{\sin a}-{\sin b}\vert \leq \vert{a}-{b}\vert$,     (b) $\vert{\arctan a}-{\arctan b}\vert \leq \vert{a}-{b}\vert$.

4.

試證若多項式 $p(x)=a_{0}x^{n}+a_{1}x^{n-1}+ \cdots +a_{n}$的所有根都是實數, 則其逐次導函數 p'(x), p''(x), $\cdots$, p⁽ⁿ⁾ 也僅有實根.

5.

Suppose that $a_{0}+a_{1}+a_{2}+ \ldots +a_{n}=0$. Show that the equation

$$a_{0}+2a_{1}x+3a_{2}x^{2}+ \ldots +(n+1)a_{n}x^{n}=0$$

has at least one root in (0,1)

6.

Let f(x)=1-x^6/7. Then f(-1)=f(1)=0, but $f'(x)\neq{0}$ for all $x\in{(-1,1)}$. Does this ocntradict Roll's theorem？

7.

試計算下列各極限:
(a) $$\lim_{x \rightarrow {\pi /2}}\frac{\tan 5x}{\tan x}$$,     (b) $$\lim_{x \rightarrow 0}\frac{x-\sin x}{x^{3}}$$,     (c) $$\lim_{x \rightarrow 0}(e^{x}-1)\cot x$$,
(d) $$\lim_{x \rightarrow \infty}x^{1/x}$$,     (e) $$\lim_{x \rightarrow 0}{\frac{1}{x}-\frac{1}{\sin x}}$$,     (f) $$\lim_{x \rightarrow 0}{{(1+3x)}^{1/x}}$$,
(g) $$\lim_{x \rightarrow \infty}(\ln{x})2/\sqrt{x}$$,     (h) $$\lim_{x \rightarrow 0^{+}}x^{1/3}(\ln{x})^{2}$$,     (i) $$\lim_{x \rightarrow 0}\frac{x-\sin x}{(e^{x}-1)^{2}}$$,
(j) $$\lim_{x \rightarrow \infty}\frac{\ln{\ln{x}}}{\ln{x-\ln{x}}}$$.

8.

設 $\vert x\vert\geq 1$. 試證

$$2{\arctan x}+\arcsin{\frac{2x}{1+x^{2}}}={\pi}\mbox{sgn}x.$$

註. $\mbox{sgn}x$ 讀作 signum of x, 其定義如下:

$$\mbox{sgn}x =\left\{ \begin{array}{ll} -1 &\mbox{若 $ x < 0 ... ...$ x = 0 $, } \\ 1 &\mbox{若 $ x > 0 $. } \end{array}\right.$$

9.

L'Hopital 定理之逆不真之例: 令

$$f(x)=x^{2}\sin \frac{1}{x}, \phi(x) = x,$$

試證

$$\lim_{x \rightarrow 0}\frac{f(x)}{\phi(x)}= \lim_{x \rightarrow 0}{x\sin\frac {1}{x}}=0,$$

但 $$\frac{f'(x)}{\phi'(x)}$$ 當 $x \rightarrow 0$ 時無極限.

10.

設 f(x) 及 g(x) 滿足下列條件:
(a) f(x), f'(x), $\ldots$, f^(n-1)(x), g(x), g'(x), $\ldots$, g^(n-1)(x) 均在 [a,b] 中連續.
(b) f⁽ⁿ⁾(x) 和 g⁽ⁿ⁾(x) 均在 (a,b) 中存在.
(c) [a,b] 中有 n+1 個點 $x_{0} < x_{1} < \cdots < x_{n}$使 f(x_i)=g(x_i), $i = 0,1,2, \ldots ,n$.
試證 (a,b) 中必有一點 $\xi$ 使 $f^{(n)}(\xi)=g^{(n)}(\xi)$.
註. n=1 時即為 Rolle's theorem.

11.

設 f(x), g(x), h(x) 均為 [a,b] 上的連續函數, 且均在 (a,b) 上可微分. 試証 (a,b) 中必有一數 $\xi$ 使

$$\left \vert \begin{array}{ccc} f^{'}(\xi)& g^{'}(\xi)& h^{'}(... ...g(a)& h(a) \\ f(b)& g(b)& h(b) \end{array} \right \vert = 0.$$

註. h(x)=1 時回到 Cauchy 的均值定理.

12.

Let f(x) and g(x) be continuous functions on [a,b]with $g(x)\geq{0}$ for all $x\in{[a,b]}$. Prove that there is a number $\xi$ in (a,b) such that

$$\int^b_a{f(x)g(x)} \, dx =f(\xi)\int^b_a{g(x)} \, dx$$

註. 本結果叫作 the weighted mean-value theorem for integrals 或第二均值定理. g(x)=1 時回到積分學的均值定理.