Find the limit of the trigonometry function

Evaluate the following limit of trigonometric function and key in the right answer. (round number)

以上函数的极限是以分数 $$\frac{A}{B}$$ 表示. 

请问A+B 是多少。 

Evaluate the limits of the following function and chose the right answer.

Evaluate the limit of the function and chose the correct answer.

Evaluate the limit of the series and chose the correct answer.

Given that $$f\left(x\right)=\left(\frac{x^2-3}{x^2+3}\right)^{\ 4}$$  , find the value of  $$f\ '\left(2\right)$$  

Given that  $$f\left(x\right)=\frac{2}{2-\cos\ x}$$  , find the value of  $$f\ '\ \left(\frac{\pi}{2}\right)$$  

Given $$f\left(x\right)=\sec\left(x^2\right)$$  , find the value of  $$f\ '\ \left(\sqrt{\frac{\pi}{4}}\right)$$  .

Given that $$f\left(x\right)=\frac{6}{x^{2\ }}$$ , find the value of    $$f\ ''\left(4\right)$$  

 Given $$s=\frac{t}{2t+1}$$ , find the velocity       (  $$v=\frac{ds}{dt}$$  ) when  $$t=3$$  .  

 Given that $$f\left(x\right)=\sqrt{x+\sqrt{x^3-1}}$$ , find the value of  $$f'\left(1\right)$$  

如图所示，OACB是一矩形。  $$AE:EC=2:3$$  且 $$\overrightarrow{OA=}\overline{a}$$  ,  $$\overrightarrow{OB}\ =\overline{b}$$  , 以  $$\overline{a}\$$  和  $$\overline{b}$$  表达  $$\overrightarrow{BE}$$ 。

如图所示，OACB 是一正方形， 且 $$AE:EC=2:1$$  而  $$BF:FE\ =\ 1:1$$  。若 $$\overrightarrow{OA}=\overline{a}$$ ，  $$\overrightarrow{OB}=\overline{b}$$  ， 以  $$\overline{a}$$  和  $$\overline{b}$$  表示  $$\overrightarrow{OF}$$  。

OABC is a tetrahedron (四面体）. M is midpoint of AB, N is midpoint of OC.  Given $$\overrightarrow{OA}\ =\overline{a\ },\ \overrightarrow{OB\ }=\overline{b},\ \overrightarrow{OC}\ =\overline{c}$$  ,  express  $$\overrightarrow{MN}$$  in term of  $$\overline{a},\ \overline{b}\ and\ \overline{c}$$  .

Straight triangular prism （直三棱柱） $$ABC\ -A_1B_1C_1$$ . Given that  $$\overrightarrow{CA}=\overline{a}$$  ,  $$\overrightarrow{CB}=\overline{b}$$  ,  $$\overrightarrow{CC_1}=\overline{c}$$  , express  $$A_1B$$  in term of  $$\overline{a},\ \overline{b},\ and\ \ \overline{c}$$  .

Given that $$\left|a\right|=3\ and\ \left|b\right|=4$$  , and the angle between these two vectors is  $$60\degree$$  . Find the value of  $$\left(2\overline{a}+\overline{b}\right)\left(\overline{a}-3\overline{b}\right)$$  .

Simplify the vectors $$\overrightarrow{CD}-\overrightarrow{AD}+\overrightarrow{BC}-\overrightarrow{BA}$$  

Given that $$\left|a\right|=6,\ \left|b\right|=4$$  and the angle between  $$\overline{a}\ and\ \overline{b}$$  is  $$37\degree$$  , find the value of  $$\overline{a}+\overline{b}$$  . (Hint : Cosine rules)

Given $$\overline{p}=3i-5j$$ ,   $$\overline{q}=-i+4j$$  , find  $$\overline{r}$$  if  $$4\overline{p}-3\overline{r}=7\overline{q}$$  .

If  $$\overline{a}\ =\ \frac{2}{3}\overline{b}$$  and  $$\overline{a}=\frac{1}{2}\overline{c}$$  , find possible value of  m and n such that  $$m\overline{c}\ +\ n\overline{b}\ =0$$ .

Given that position vector $$A=\overline{x}+2\overline{y}$$    $$B=3\overline{x}-\overline{y}$$  ,  $$C=4\overline{x}+k\overline{y}$$  ,and   $$\overline{x}\cdot\overline{y}=0$$  .  If  $$\overrightarrow{AB}$$  perpendicular  $$\perp$$  with  $$\overrightarrow{BC}$$  and  $$\left|\overline{x}\right|=\left|\overline{y}\right|$$  , find the value of  $$k$$  .

Roll three fair dices at the same time, the probability of getting sum of 10 is ?

A bulb factory produce and pack their products into 10 pieces per box. QC department inspect the bulb by checking 4 bulbs randomly for every box. If 2 or more defects bulb being inspected, the box will be rejected. One of the box contains of 3 defect bulbs, calculate the probability of the box being rejected.

In a factory, machine A,B,C produce same products in ratio 5:3:2. According to past records , machine A:B:C produce defect rate product are 5% : 3% : 4%.

One defect products is draw among the defect products and calculate the probability of the defect product belong to machine A.

Black and red dices are rolled continuously. Find the conditional probability of obtaining a sum greater than 9, given that black dice resulted in a 5.
黑色和红色骰子分别掷出，计算条件概率两骰子的点数大于9，如果已知黑色骰子的点数为5.

A number is chosen at random from the set,  $$S=\left\{1,2,3,4,5,6,7,8,9,10\right\}$$  . Find the probability that the chosen number is an odd number or a prime number. （Prime numbers=2，3，5，7）

There are 6 black balls, 5 red balls and 4 blue balls in a bag. Two balls are drawn at random from the bag. Find the probability that both drawn balls are of same colour.

Toss an unfair coin to get head is  $$\frac{1}{3}$$  , and roll an unfair dice to get odd number is  $$\frac{3}{5}$$  . Calculate the probability to get result of tail and even number by toss the unfair coin and roll the unfair dice at the same time.

12 males and 12 females attend an competition that need pairing. Find the probability of at least 1 pair forming by 1 male and 1 female.

12男12女参加一个比赛，并需要组两人的队伍。计算至少有一队由一男一女组成的概率。

There are 8 red marbles, 4 blue marbles and 3 white marbles in a box. Three marbles are drawn from the box, one after another, without replacement. Find the probability that at least one of the marbles is blue.

The probability of A pass the add maths is  $$\frac{2}{3}$$ , while B to pass the add maths is  $$\frac{5}{7}$$   . Find the probability that only one of then passes the add maths.