Mister Exam

Lang:

Limit of the function (1+4/x)^(3*x)

You have entered [src]

            3*x
     /    4\   
 lim |1 + -|   
x->oo\    x/

\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3 x}

Limit((1 + 4/x)^(3*x), x, oo, dir='-')

Detail solution

Let's take the limit

\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3 x}

transform
do replacement

u = \frac{x}{4}

then

\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3 x}

=
=

\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{12 u}

\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{12 u}

\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{12}

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)^{12} = e^{12}

The final answer:

\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3 x} = e^{12}

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]

 12
e

e^{12}

Expand and simplify

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3 x} = e^{12}

\lim_{x \to 0^-} \left(1 + \frac{4}{x}\right)^{3 x} = 1

\lim_{x \to 0^+} \left(1 + \frac{4}{x}\right)^{3 x} = 1

\lim_{x \to 1^-} \left(1 + \frac{4}{x}\right)^{3 x} = 125

\lim_{x \to 1^+} \left(1 + \frac{4}{x}\right)^{3 x} = 125

\lim_{x \to -\infty} \left(1 + \frac{4}{x}\right)^{3 x} = e^{12}

The graph