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(1+n)/(2+n)

Limit of the function (1+n)/(2+n)

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     /1 + n\
 lim |-----|
n->oo\2 + n/
limn(n+1n+2)\lim_{n \to \infty}\left(\frac{n + 1}{n + 2}\right)
Limit((1 + n)/(2 + n), n, oo, dir='-')
Detail solution
Let's take the limit
limn(n+1n+2)\lim_{n \to \infty}\left(\frac{n + 1}{n + 2}\right)
Let's divide numerator and denominator by n:
limn(n+1n+2)\lim_{n \to \infty}\left(\frac{n + 1}{n + 2}\right) =
limn(1+1n1+2n)\lim_{n \to \infty}\left(\frac{1 + \frac{1}{n}}{1 + \frac{2}{n}}\right)
Do Replacement
u=1nu = \frac{1}{n}
then
limn(1+1n1+2n)=limu0+(u+12u+1)\lim_{n \to \infty}\left(\frac{1 + \frac{1}{n}}{1 + \frac{2}{n}}\right) = \lim_{u \to 0^+}\left(\frac{u + 1}{2 u + 1}\right)
=
102+1=1\frac{1}{0 \cdot 2 + 1} = 1

The final answer:
limn(n+1n+2)=1\lim_{n \to \infty}\left(\frac{n + 1}{n + 2}\right) = 1
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
limn(n+1)=\lim_{n \to \infty}\left(n + 1\right) = \infty
and limit for the denominator is
limn(n+2)=\lim_{n \to \infty}\left(n + 2\right) = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
limn(n+1n+2)\lim_{n \to \infty}\left(\frac{n + 1}{n + 2}\right)
=
limn(ddn(n+1)ddn(n+2))\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(n + 1\right)}{\frac{d}{d n} \left(n + 2\right)}\right)
=
limn1\lim_{n \to \infty} 1
=
limn1\lim_{n \to \infty} 1
=
11
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
The graph
02468-8-6-4-2-1010-2525
Other limits n→0, -oo, +oo, 1
limn(n+1n+2)=1\lim_{n \to \infty}\left(\frac{n + 1}{n + 2}\right) = 1
limn0(n+1n+2)=12\lim_{n \to 0^-}\left(\frac{n + 1}{n + 2}\right) = \frac{1}{2}
More at n→0 from the left
limn0+(n+1n+2)=12\lim_{n \to 0^+}\left(\frac{n + 1}{n + 2}\right) = \frac{1}{2}
More at n→0 from the right
limn1(n+1n+2)=23\lim_{n \to 1^-}\left(\frac{n + 1}{n + 2}\right) = \frac{2}{3}
More at n→1 from the left
limn1+(n+1n+2)=23\lim_{n \to 1^+}\left(\frac{n + 1}{n + 2}\right) = \frac{2}{3}
More at n→1 from the right
limn(n+1n+2)=1\lim_{n \to -\infty}\left(\frac{n + 1}{n + 2}\right) = 1
More at n→-oo
Rapid solution [src]
1
11
The graph
Limit of the function (1+n)/(2+n)