What is the limit of #(1+x)^(1/x) # as x approaches 0? Calculus Limits Determining Limits Algebraically 1 Answer Cesareo R. Dec 19, 2016 #e# Explanation: From the binomial expansion #(1+x)^n = 1 + nx + (n(n-1))/(2!)x^2+(n(n-1)(n-2))/(3!)x^3+ cdots +# we have #(1+x)^(1/x) = 1 + 1/x x + (1/x(1/x-1))/(2!)x^2+(1/x(1/x-1)(1/x-2))/(3!)x^3+ cdots +# #=1+1+(1(1-x))/(2!)+(1(1-x)(1-2x))/(3!)+cdots+# so #lim_(x->0)(1+x)^(1/x) =lim_(x->0)1+1+(1(1-x))/(2!)+(1(1-x)(1-2x))/(3!)+cdots+=sum_(k=0)^oo1/(k!) = e# Answer link Related questions How do you find the limit #lim_(x->5)(x^2-6x+5)/(x^2-25)# ? How do you find the limit #lim_(x->3^+)|3-x|/(x^2-2x-3)# ? How do you find the limit #lim_(x->4)(x^3-64)/(x^2-8x+16)# ? How do you find the limit #lim_(x->2)(x^2+x-6)/(x-2)# ? How do you find the limit #lim_(x->-4)(x^2+5x+4)/(x^2+3x-4)# ? How do you find the limit #lim_(t->-3)(t^2-9)/(2t^2+7t+3)# ? How do you find the limit #lim_(h->0)((4+h)^2-16)/h# ? How do you find the limit #lim_(h->0)((2+h)^3-8)/h# ? How do you find the limit #lim_(x->9)(9-x)/(3-sqrt(x))# ? How do you find the limit #lim_(h->0)(sqrt(1+h)-1)/h# ? See all questions in Determining Limits Algebraically Impact of this question 102722 views around the world You can reuse this answer Creative Commons License