이 한계를 어떻게 해결합니까?

대답:

# e ^ a * (a / 2) * (1 - a) #

설명:

# ""Taylor 시리즈를 사용하고 "#" # "제한"x-> 0 "."#

# x ^ y = exp (y * ln (x)) #

# => (1 + x) ^ y = exp (y * ln (1 + x)) #

# "및"ln (1 + x) = x - x ^ 2 / 2 + x ^ 3 / 3 - … #

# "및"exp (x) = 1 + x + x ^ 2 / 2 + x ^ 3 / 6 + x ^ 4 / 24 + … #

#"그래서"#

#exp (y * ln (1 + x)) = exp (y * (x-x ^ 2 / 2 + …)) #

# (1 + x) ^ (a / x) = exp ((a / x) * ln (1 + x)) #

# = exp ((a / x) * (x - x ^ 2 / 2 + x ^ 3 / 3 - …)) #

# = exp (a - a * x / 2 + a * x ^ 2 / 3 - …) #

# (1 + ax) ^ (1 / x) = exp ((1 / x) * ln (1 + ax)) #

# = exp ((1 / x) * (ax - (ax) ^ 2 / 2 + (ax) ^ 3 / 3 - …)) #

# = exp (a - a ^ 2 * x / 2 + a ^ 3 * x ^ 2 / 3 - …) #

= (1 + ax) ^ (1 / x) - (1 + x) ^ (a / x) #

# ~~exp (a-a * 2 / x + 2 + …) - exp (a-a * x / 2 + …) #

# a ~ exp (a) / exp (a ^ 2 * x / 2) - exp (a) / exp (a * x / 2) #

# = exp (a) (exp (-a * 2 * x / 2) - exp (-a * x / 2)) #

# ~ ~ exp (a) (1 - a ^ 2 * x / 2 - 1 + a * x / 2) #

# = exp (a) ((x / 2) (a - a ^ 2)) #

# (1 + ax) ^ (1 / x) - (1 + x) ^ (a / x)) / x #

# ~~exp (a) ((1/2) (a-a ^ 2)) #

# = e ^ a * (a / 2) * (1 - a) #