Mister Exam

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Limit of the function:
Limit of (1+x^2-4*x)/(1+2*x)
Limit of (4+x^2)/(-6+2*x)
Limit of (-x+tan(x))/(x+2*sin(x))
Limit of (sqrt(5+x)-sqrt(10))/(-15+x^2-2*x)
Identical expressions
(one - one /(three *x))^(six *x)
(1 minus 1 divide by (3 multiply by x)) to the power of (6 multiply by x)
(one minus one divide by (three multiply by x)) to the power of (six multiply by x)
(1-1/(3*x))(6*x)
1-1/3*x6*x
(1-1/(3x))^(6x)
(1-1/(3x))(6x)
1-1/3x6x
1-1/3x^6x
(1-1 divide by (3*x))^(6*x)
Similar expressions
(1+1/(3*x))^(6*x)

Limit of the function (1-1/(3x))^(6x)

You have entered [src]

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

\lim_{x \to \infty} \left(1 - \frac{1}{3 x}\right)^{6 x}

Limit((1 - 1/(3*x))^(6*x), x, oo, dir='-')

Detail solution

Let's take the limit

\lim_{x \to \infty} \left(1 - \frac{1}{3 x}\right)^{6 x}

transform
do replacement

u = \frac{x}{- \frac{1}{3}}

then

\lim_{x \to \infty} \left(1 - \frac{1}{3 x}\right)^{6 x}

=
=

\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{- 2 u}

\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{- 2 u}

\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-2}

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)^{-2} = e^{-2}

The final answer:

\lim_{x \to \infty} \left(1 - \frac{1}{3 x}\right)^{6 x} = e^{-2}

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]

 -2
e

e^{-2}

Expand and simplify

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty} \left(1 - \frac{1}{3 x}\right)^{6 x} = e^{-2}

\lim_{x \to 0^-} \left(1 - \frac{1}{3 x}\right)^{6 x} = 1

\lim_{x \to 0^+} \left(1 - \frac{1}{3 x}\right)^{6 x} = 1

\lim_{x \to 1^-} \left(1 - \frac{1}{3 x}\right)^{6 x} = \frac{64}{729}

\lim_{x \to 1^+} \left(1 - \frac{1}{3 x}\right)^{6 x} = \frac{64}{729}

\lim_{x \to -\infty} \left(1 - \frac{1}{3 x}\right)^{6 x} = e^{-2}

The graph

Limit of the function (1-1/(3*x))^(6*x)