I have a question which relate adjoint space. That is: ” Is $C^*[0,1]$ linear isomorphism with $C[0,1]$?” $C[0,1]$ is space of continous function in $[0,1]$. If we choose inner product of $C[0,1]$ is $<,>: C[0,1] \longrightarrow\Bbb{R}$, by $=\int_0^1 f(x)g(x)dx$ then $C^*[0,1]$ $=$ $C[0,1]$ (linear isomorphism)?

Problem of division boy and girl set

TST Lam Dong 2010

Make $n$ boys and $n$ girls a line. Divide the line by two parts such that number of boys and number of girls are equivalent. Let $A$ be number of cases which cannot divide, let $B$ be number of cases which can divide by one way. Prove that $B = 2A$.

Solution. One way division the line is choosing $k$ boys and $k$ girls. The line has two part, left part and right part. And two ways division are equivalent if their left part has same of number boys and girls.

With the only way division, (suppose $k$ boys and $k$ girls in left part), we change a boy in left part with a girl in right part then we attain a impossible way. And we change a girl in left part with a boy in right part then we attain a impossible way too.

So we have correspondence of one-one: the one way division and two impossible way division, this implies $B = 2A$. QEA

Bài tập về khử dạng vô định

Khi tính giới hạn hàm số, ta sẽ gặp những bài toán có dạng $\infty - \infty, \dfrac{\infty}{\infty}, \dfrac{0}{0}, 0.\infty, 1^{\infty}$. Đó là các dạng vô định. Để làm các kiểu bài tập này, các em phải biết cách khử chúng.

Yêu cầu: Nắm vững phần giới hạn hàm số, giới hạn một phía + các hằng đẳng thức.

Bài 1.

1. $\underset{x\to 1}{\lim}\dfrac{-2{{x}^{2}}+x+1}{{{x}^{2}}-4x+3}$.
2. $\underset{x\to 2}{\lim}\dfrac{\sqrt{2x-1}-\sqrt{x+1}}{{{x}^{2}}-3x+2}$.
3. $\underset{x\to 0}{\lim }\dfrac{\sqrt[3]{1+x}-\sqrt[3]{1-x}}{x}$.
4. $\underset{x\to 1}{\lim }\dfrac{\sqrt[3]{x+7}-\sqrt{x+3}}{{{x}^{2}}-3x+2}$.

Bài 2.

1. $\underset{x\to 2}{\lim }\dfrac{\sqrt{x+7}-3}{{{x}^{2}}-4}$.
2. $\underset{x\to -1}{ \lim }\dfrac{\sqrt{{{x}^{2}}+3}-2}{\sqrt[3]{x}+1}$.
3. $\underset{x\to -\infty }{\lim }\dfrac{\sqrt{{{x}^{2}}+2x+3}+2x}{\sqrt{4{{x}^{2}}+2}-x+1}$.

Một số dạng vô định có chứa lượng giác, ta chú ý giới hạn sau:

$\lim\limits_{x \to 0}\dfrac{\sin x}{x}=1$.

Bài 3.

1. $\lim\limits_{x \to o}\dfrac{1 - \cos x}{x^2}$.
2. $\lim\limits_{x \to 0}\dfrac{\sin 2x}{\sin 5x}$.
3. $\lim\limits_{x \to 0}\dfrac{\sqrt{1+x^2} - \cos x}{x^2}$.