Mister Exam

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Limit of the function (1+5/n)^n

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            n
     /    5\ 
 lim |1 + -| 
n->oo\    n/

\lim_{n \to \infty} \left(1 + \frac{5}{n}\right)^{n}

Limit((1 + 5/n)^n, n, oo, dir='-')

Detail solution

Let's take the limit

\lim_{n \to \infty} \left(1 + \frac{5}{n}\right)^{n}

transform
do replacement

u = \frac{n}{5}

then

\lim_{n \to \infty} \left(1 + \frac{5}{n}\right)^{n}

=
=

\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{5 u}

\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{5 u}

\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{5}

The limit

\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}

is second remarkable limit, is equal to e ~ 2.718281828459045
then

\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{5} = e^{5}

The final answer:

\lim_{n \to \infty} \left(1 + \frac{5}{n}\right)^{n} = e^{5}

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

Plot the graph

Other limits n→0, -oo, +oo, 1

\lim_{n \to \infty} \left(1 + \frac{5}{n}\right)^{n} = e^{5}

\lim_{n \to 0^-} \left(1 + \frac{5}{n}\right)^{n} = 1

\lim_{n \to 0^+} \left(1 + \frac{5}{n}\right)^{n} = 1

\lim_{n \to 1^-} \left(1 + \frac{5}{n}\right)^{n} = 6

\lim_{n \to 1^+} \left(1 + \frac{5}{n}\right)^{n} = 6

\lim_{n \to -\infty} \left(1 + \frac{5}{n}\right)^{n} = e^{5}

Rapid solution [src]

5
e

e^{5}

Expand and simplify

The graph