Question

极限计算题x趋向0 (1+1/x)^x=

Answer 1

lim x->0 (1+1/x)^x=lim x->0 e^[x*ln(1+1/x)]

当x->0-时,1+1/x->负无穷,ln(1+1/x)无意义;

当x->0+时,1+1/x->正无穷,所以ln(1+1/x)->正无穷,由洛必达法则知x*ln(1+1/x)->0,此时lim x->0+ (1+1/x)^x=e^[lim x->0+ x*ln(1+1/x)]=e^0=1.[用到了函数的连续性知识]

综上,只有x->0+时,有lim x->0+ (1+1/x)^x=1.