To determine
To show:
that limn→∞(1+xn)n=ex for any x>0
Answer
limn→∞(1+xn)n=ex for any x>0
Explanation
Proof:
Consider the limit:
limn→∞(1+xn)n
Let m=nx
So, n=xm
This implies, as n→∞,m→∞.
Therefore,
limn→∞(1+xn)n=limm→∞(1+1m)mx=[limm→∞(1+1m)m]x=ex
Since, limn→∞(1+1n)n=e. This implies:
limn→∞(1+xn)n=[limm→∞(1+1m)m]x=ex
Thus, limn→∞(1+xn)n=ex
Conclusion:
Hence, limn→∞(1+xn)n=ex for any x>0