why does the limit as x goes to infinity for (1+2/x)^x is e^2..can someone sow the steps..thanks
To understand why the limit of (1 + 2/x)^x as x approaches infinity is e^2 (approximately 7.389), we can follow these steps:
Step 1: Rewrite the expression using exponential notation.
(1 + 2/x)^x = [(1 + 2/x)^(x/2)]^2
Step 2: Substitute a variable, such as n, for x/2.
Let n = x/2, which implies x = 2n.
Step 3: Rewrite the expression again using the new variable n.
([(1 + 2/x)^(x/2)])^2 = [(1 + 2/(2n))^(2n)]^2
Simplifying, we get [(1 + 1/n)^n]^2 = (1 + 1/n)^(2n)
Step 4: Apply the definition of e.
The limit of (1 + 1/n)^n as n approaches infinity equals e. Therefore, we need to calculate the limit of [(1 + 1/n)^(2n)] as n approaches infinity.
Step 5: Rewrite the expression using exponential notation.
[(1 + 1/n)^(2n)] = ([(1 + 1/n)^(n)])^2
Step 6: Recall that the limit of (1 + 1/n)^n as n approaches infinity is e.
So, [(1 + 1/n)^(n)] approaches e.
Therefore, [(1 + 1/n)^(2n)] approaches e^2.
Step 7: Conclude the original limit.
Since [(1 + 1/n)^(2n)] approaches e^2, the limit of [(1 + 2/x)^(x/2)]^2 as x approaches infinity is also e^2.
Hence, the limit as x approaches infinity of (1 + 2/x)^x is e^2.