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((1+n)/n)^n
Limit of the function/ ((1+n)/n)^n

Limit of the function ((1+n)/n)^n

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            n
     /1 + n\ 
 lim |-----| 
n->oo\  n  /
lim⁡n→∞(n+1n)n\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n} 
Limit(((1 + n)/n)^n, n, oo, dir='-')
Detail solution
Let's take the limit
lim⁡n→∞(n+1n)n\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n}
transform
lim⁡n→∞(n+1n)n\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n}
=
lim⁡n→∞(n+1n)n\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n}
=
lim⁡n→∞(nn+1n)n\lim_{n \to \infty} \left(\frac{n}{n} + \frac{1}{n}\right)^{n}
=
lim⁡n→∞(1+1n)n\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n}
=
do replacement
u=n1u = \frac{n}{1}
then
lim⁡n→∞(1+1n)n\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n} =
=
lim⁡u→∞(1+1u)u\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}
=
lim⁡u→∞(1+1u)u\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}
=
((lim⁡u→∞(1+1u)u))\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)
The limit
lim⁡u→∞(1+1u)u\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}
is second remarkable limit, is equal to e ~ 2.718281828459045
then
((lim⁡u→∞(1+1u)u))=e\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right) = e

The final answer:
lim⁡n→∞(n+1n)n=e\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n} = e
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
02468-8-6-4-2-1010020
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Rapid solution [src] 
E
ee
Expand and simplify
The graph
Limit of the function ((1+n)/n)^n
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