求极限 lim（2^x+3^x -2）/x 当X趋近于0

由洛必达法则,lim (x→0) (2^x +3^x -2) /x = lim (x→0) [(2^x) (ln 2)+(3^x) (ln 3) ] /1= ln6。= = = = = = = = =如果没学洛必达法则,但学了等价无穷小量,见解法2。解法2：因为 lim (t→0) (e^t -1) /t =1,令 t =x (ln 2),则 x = t /(ln2)。所以 t→0 时,x→0。所以 lim (x→0) (2^x -1) / [ x (ln2) ] =1,所以 lim (x→0) (2^x -1) /x =ln2。同理,lim (x→0) (3^x -1) /x =ln3。所以 lim (x→0) (2^x +3^x -2) /x = lim (x→0) (2^x -1) /x +lim (x→0) (3^x -1) /x= ln6。= = = = = = = = =解法3：（导数的定义）令 f(t) =2^t,则 f'(t) =(2^t) (ln2),所以 f'(0) =ln2,即 lim (x→0) (2^x -2^0) /(x -0) =ln2,即 lim (x→0) (2^x -1) /x =ln2。同理,lim (x→0) (3^x -1) /x =ln3。所以 lim (x→0) (2^x +3^x -2) /x = lim (x→0) (2^x -1) /x +lim (x→0) (3^x -1) /x= ln6。