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Limit of the function (1+3/x)^(3*x)

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            3*x
     /    3\   
 lim |1 + -|   
x->oo\    x/

\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{3 x}

Limit((1 + 3/x)^(3*x), x, oo, dir='-')

Detail solution

Let's take the limit

\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{3 x}

transform
do replacement

u = \frac{x}{3}

then

\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{3 x}

=
=

\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{9 u}

\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{9 u}

\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{9}

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)^{9} = e^{9}

The final answer:

\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{3 x} = e^{9}

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

Rapid solution [src]

9
e

e^{9}

Expand and simplify

The graph