How do you find the limit of #(e^x + sin(x) + x - 1)/(e^x - 1)# as x approaches 0? Calculus Limits Determining Limits Algebraically 1 Answer Cesareo R. Jun 2, 2016 #lim_{x->0}(e^x+sin(x)+x-1)(e^x-1) = 3# Explanation: #f(x)=(e^x+sin(x)+x-1)(e^x-1) = (e^x-1)/(e^x-1)+(sin(x)+x)/(e^e-1)# so #f(x)=1+(x(sin(x)/x+1))/(x(e^x-1)/x) = 1 + (sin(x)/x+1)/((e^x-1)/x)# Here #lim_{x->0}sin(x)/x=1# and #lim_{x->0}(e^x-1)/x=1# Putting all together #lim_{x->0}f(x)=1+2/1=3# 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 2108 views around the world You can reuse this answer Creative Commons License