微積分:指數函數

(1)

Consider _µ

1 + 1



¶

 where  is a positive integral. Then

lim →+∞ µ 1 + 1  ¶ =  _{≈ 27182818284}

Theorem 50 If  is any positive rational number, then

(1) lim →+∞ ³ 1 +   ´  = ; (2) lim →+∞ ³ 1 +   ´ = 

Theorem 51 Let  and  be any numbers.

1. _ _{= }+_; 2. _ = −; 3. () = ; 4. 0 _{= 1;} 5. −₌ 1 ; 6. If  =  then  =  56

(2)

Exponetial Functions

Definition 22 An exponential function is a function that can be expressed

form

 () = 

where   0 and  6= 0

Properties 1. The domain of  () = 

is (−∞ ∞)  and the range is

(0_{∞) }

2. The function  () =  _{is continuous, increasing, and one-to-one on}

R

3. The graph of  () =  _{is concave upward on its entire domain.}

4. lim_→−∞ _{= 0} _{and lim}

→+∞ = +∞ 5 2.5 0 -2.5 -5 125 100 75 50 25 0 x y x y

5. lim_→∞− _{= 0} _{and lim}

→−∞− = +∞ 5 2.5 0 -2.5 -5 125 100 75 50 25 0 x y x y 57