Lecture 3 : Monotone Sequence and Limit theorem [ Section 3.2 : Limit Theorems on Sequences ]
Hence, .
Using binomial theorem, we get .
Thus, .
Hence, by the Sandwich Theorem, as .
(ii)
For x > 0, the sequence is convergent and . To see this, let us first suppose that
x > 1.Then, for every . Let .
Once again, for every . .
Hence, and by Sandwitch theorem, , i.e., Next, suppose .
Then for every n. Let .
Once again, for every , .
.
.
.
as 
.
is convergent and 
. To see this, let us first suppose that 
for every 
. Let
. 
. 
and by Sandwitch theorem, 
, i.e., 
Next, suppose 
. 
for every n. Let
. 
.
