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((1+x)/(2+x))^(1+x)
Limit of the function/ ((1+x)/(2+x))^(1+x)

Limit of the function ((1+x)/(2+x))^(1+x)

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            1 + x
     /1 + x\     
 lim |-----|     
x->oo\2 + x/
lim⁡x→∞(x+1x+2)x+1\lim_{x \to \infty} \left(\frac{x + 1}{x + 2}\right)^{x + 1} 
Limit(((1 + x)/(2 + x))^(1 + x), x, oo, dir='-')
Detail solution
Let's take the limit
lim⁡x→∞(x+1x+2)x+1\lim_{x \to \infty} \left(\frac{x + 1}{x + 2}\right)^{x + 1}
transform
lim⁡x→∞(x+1x+2)x+1\lim_{x \to \infty} \left(\frac{x + 1}{x + 2}\right)^{x + 1}
=
lim⁡x→∞((x+2)−1x+2)x+1\lim_{x \to \infty} \left(\frac{\left(x + 2\right) - 1}{x + 2}\right)^{x + 1}
=
lim⁡x→∞(−1x+2+x+2x+2)x+1\lim_{x \to \infty} \left(- \frac{1}{x + 2} + \frac{x + 2}{x + 2}\right)^{x + 1}
=
lim⁡x→∞(1−1x+2)x+1\lim_{x \to \infty} \left(1 - \frac{1}{x + 2}\right)^{x + 1}
=
do replacement
u=x+2−1u = \frac{x + 2}{-1}
then
lim⁡x→∞(1−1x+2)x+1\lim_{x \to \infty} \left(1 - \frac{1}{x + 2}\right)^{x + 1} =
=
lim⁡u→∞(1+1u)−u−1\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{- u - 1}
=
lim⁡u→∞((1+1u)−u1+1u)\lim_{u \to \infty}\left(\frac{\left(1 + \frac{1}{u}\right)^{- u}}{1 + \frac{1}{u}}\right)
=
lim⁡u→∞11+1ulim⁡u→∞(1+1u)−u\lim_{u \to \infty} \frac{1}{1 + \frac{1}{u}} \lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{- u}
=
lim⁡u→∞(1+1u)−u\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{- u}
=
((lim⁡u→∞(1+1u)u))−1\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-1}
The limit
lim⁡u→∞(1+1u)u\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}
is second remarkable limit, is equal to e ~ 2.718281828459045
then
((lim⁡u→∞(1+1u)u))−1=e−1\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-1} = e^{-1}

The final answer:
lim⁡x→∞(x+1x+2)x+1=e−1\lim_{x \to \infty} \left(\frac{x + 1}{x + 2}\right)^{x + 1} = e^{-1}
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
02468-8-6-4-2-10100.01.0
Plot the graph
Rapid solution [src] 
 -1
e
e−1e^{-1}
Expand and simplify
The graph
Limit of the function ((1+x)/(2+x))^(1+x)
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