Could you please prove that #lim_(x rarr 0) sin(x)/x = 1# using the formal definition of limits?
1 Answer
Explanation:
Recall:

Error Bound for Alternating Series
Let#s=sum_(n=0)^(infty)(1)^nb_n# , where#b_n>0# and#b_n# is monotonically decreasing, and let
#s_n=sum_(i=0)^n(1)^n b_n# .
The error for approximating the sum#s# using the partial sum#s_n# is bounded by#b_(n+1)# , that is,#ss_n leq b_(n+1)# 
Power Series of sin x
#sin x =xx^3/(3!)+x^5/(5!)x^7/(7!)+cdots#
So, the error bound for estimating
Let us now prove the limit.
By Error Bound for Alternating Seires,
Hence,