lim┬(x→∞)⁡〖(1/x+2^(1/x) )^x 〗求极限,

lim┬(x→∞)⁡〖(1/x+2^(1/x) )^x 〗求极限,
lim┬(x→∞)⁡〖(1/x+2^(1/x) )^x 〗求极限,

lim┬(x→∞)⁡〖(1/x+2^(1/x) )^x 〗求极限,
lim(x→∞) [1/x +2^(1/x)] ^x 令 t=1/x
原式= lim(t->0) (t + 2^t) ^ (1/t)
= e^ lim(t->0) ln( t+2^t) / t
lim(t->0) ln(t+2^t) / t
= lim(t->0) ln(1+ t + 2^t -1) / t 等价无穷小代换 u = t+2^t -1 -> 0,ln(1+u) u
= lim(t->0) ( t+ 2^t -1) / t 等价无穷小代换 2^t - 1 t * ln2
= lim(t->0) ( t+ ln2* t) / t = 1 + ln2
原式 = e^(1+ln2) = 2e