求极限 lim（2^x+3^x -2）⼀x 当X趋近于0

解：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.

负无穷大