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