Show that limit as n approaches infinity of (1+x/n)^n=e^x for any x>0...

Should i use the formula e= lim as x->0 (1+x)^(1/x)

or

e= lim as x->infinity (1+1/n)^n

Am i able to substitute in x/n for x? and then say that

e lim x ->0 (1+x/n)^(1/(x/n))

and then raise it to the xth power

ie. e^x and lim x -> (1+x/n)^(n)

Thanks for any help.. just tell me if is correct please or if i am on the right track


also,, how do i say that this is for all x>0?

To prove that the limit as n approaches infinity of (1+x/n)^n is equal to e^x for any x>0, you can use the formula e = lim as x approaches 0 (1+x)^(1/x).

First, let's confirm that the given limit e^(x)= lim as x→∞ (1+1/n)^n is a valid expression. You cannot directly substitute x/n for x because here we need x to approach infinity, not just any value. Therefore, you should use the original formula instead of the x/n substitution.

To show that the limit e^(x) = lim as n approaches infinity (1+x/n)^n is valid for any x>0, we need to prove that the two formulas are equivalent.

We start with the limit e = lim as x approaches 0 (1+x)^(1/x). Taking the natural logarithm of both sides, we get:

ln(e) = ln(lim as x approaches 0 (1+x)^(1/x))

Since ln(e) = 1, we now have:

1 = ln(lim as x approaches 0 (1+x)^(1/x))

Using the property of logarithms, we can rewrite this as:

e^(1) = lim as x approaches 0 (1+x)^(1/x)

Now, substitute x = nx in the above equation, where n is any positive integer. This gives:

e^(1) = lim as nx approaches 0 (1+nx)^(1/(nx))

Since nx still approaches 0 as n approaches infinity, we can express it as:

e^(1) = lim as n approaches infinity [(1+(x/n))^n]

Now, by comparing this expression with the original limit e^(x) = lim as n approaches infinity (1+x/n)^n, we can see that they are identical.

Therefore, you can conclude that for any x>0, the limit as n approaches infinity of (1+x/n)^n is equal to e^x.

Regarding the statement that this holds for all x>0, you can simply add this statement at the beginning or end of your proof. For example:

"Consider the limit of (1+x/n)^n as n approaches infinity. To prove that this limit is equal to e^x, we show that it is equivalent to the formula e^(x) = lim as n approaches infinity (1+x/n)^n. Since this holds for any x>0, we can conclude that for all x>0, the limit as n approaches infinity of (1+x/n)^n is equal to e^x."