Mister Exam

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Limit of the function ((1+x)/(11+x))^x

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             x
     /1 + x \ 
 lim |------| 
x->oo\11 + x/

\lim_{x \to \infty} \left(\frac{x + 1}{x + 11}\right)^{x}

Limit(((1 + x)/(11 + x))^x, x, oo, dir='-')

Detail solution

Let's take the limit

\lim_{x \to \infty} \left(\frac{x + 1}{x + 11}\right)^{x}

transform

\lim_{x \to \infty} \left(\frac{x + 1}{x + 11}\right)^{x}

\lim_{x \to \infty} \left(\frac{\left(x + 11\right) - 10}{x + 11}\right)^{x}

\lim_{x \to \infty} \left(- \frac{10}{x + 11} + \frac{x + 11}{x + 11}\right)^{x}

\lim_{x \to \infty} \left(1 - \frac{10}{x + 11}\right)^{x}

=
do replacement

u = \frac{x + 11}{-10}

then

\lim_{x \to \infty} \left(1 - \frac{10}{x + 11}\right)^{x}

=
=

\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{- 10 u - 11}

\lim_{u \to \infty}\left(\frac{\left(1 + \frac{1}{u}\right)^{- 10 u}}{\left(1 + \frac{1}{u}\right)^{11}}\right)

\lim_{u \to \infty} \frac{1}{\left(1 + \frac{1}{u}\right)^{11}} \lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{- 10 u}

\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{- 10 u}

\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-10}

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)^{-10} = e^{-10}

The final answer:

\lim_{x \to \infty} \left(\frac{x + 1}{x + 11}\right)^{x} = e^{-10}

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]

 -10
e

e^{-10}

Expand and simplify

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty} \left(\frac{x + 1}{x + 11}\right)^{x} = e^{-10}

\lim_{x \to 0^-} \left(\frac{x + 1}{x + 11}\right)^{x} = 1

\lim_{x \to 0^+} \left(\frac{x + 1}{x + 11}\right)^{x} = 1

\lim_{x \to 1^-} \left(\frac{x + 1}{x + 11}\right)^{x} = \frac{1}{6}

\lim_{x \to 1^+} \left(\frac{x + 1}{x + 11}\right)^{x} = \frac{1}{6}

\lim_{x \to -\infty} \left(\frac{x + 1}{x + 11}\right)^{x} = e^{-10}

The graph