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,#|s-s_n| leq b_(n+1)# -
Power Series of sin x
#sin x =x-x^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,