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

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

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     /n*(2 + n)\
 lim |---------|
n->oo|        2|
     \ (1 + n) /
lim⁡n→∞(n(n+2)(n+1)2)\lim_{n \to \infty}\left(\frac{n \left(n + 2\right)}{\left(n + 1\right)^{2}}\right) 
Limit((n*(2 + n))/(1 + n)^2, n, oo, dir='-')
Detail solution
Let's take the limit
lim⁡n→∞(n(n+2)(n+1)2)\lim_{n \to \infty}\left(\frac{n \left(n + 2\right)}{\left(n + 1\right)^{2}}\right)
Let's divide numerator and denominator by n^2:
lim⁡n→∞(n(n+2)(n+1)2)\lim_{n \to \infty}\left(\frac{n \left(n + 2\right)}{\left(n + 1\right)^{2}}\right) =
lim⁡n→∞(1+2n1+2n+1n2)\lim_{n \to \infty}\left(\frac{1 + \frac{2}{n}}{1 + \frac{2}{n} + \frac{1}{n^{2}}}\right)
Do Replacement
u=1nu = \frac{1}{n}
then
lim⁡n→∞(1+2n1+2n+1n2)=lim⁡u→0+(2u+1u2+2u+1)\lim_{n \to \infty}\left(\frac{1 + \frac{2}{n}}{1 + \frac{2}{n} + \frac{1}{n^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{2 u + 1}{u^{2} + 2 u + 1}\right)
=
0⋅2+102+0⋅2+1=1\frac{0 \cdot 2 + 1}{0^{2} + 0 \cdot 2 + 1} = 1

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

i.e. limit for the numerator is
lim⁡n→∞(n(n+2))=∞\lim_{n \to \infty}\left(n \left(n + 2\right)\right) = \infty
and limit for the denominator is
lim⁡n→∞(n+1)2=∞\lim_{n \to \infty} \left(n + 1\right)^{2} = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡n→∞(n(n+2)(n+1)2)\lim_{n \to \infty}\left(\frac{n \left(n + 2\right)}{\left(n + 1\right)^{2}}\right)
=
Let's transform the function under the limit a few
lim⁡n→∞(n(n+2)(n+1)2)\lim_{n \to \infty}\left(\frac{n \left(n + 2\right)}{\left(n + 1\right)^{2}}\right)
=
lim⁡n→∞(ddnn(n+2)ddn(n+1)2)\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n \left(n + 2\right)}{\frac{d}{d n} \left(n + 1\right)^{2}}\right)
=
lim⁡n→∞1\lim_{n \to \infty} 1
=
lim⁡n→∞1\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
02468-8-6-4-2-1010-200100
Plot the graph
Other limits n→0, -oo, +oo, 1
lim⁡n→∞(n(n+2)(n+1)2)=1\lim_{n \to \infty}\left(\frac{n \left(n + 2\right)}{\left(n + 1\right)^{2}}\right) = 1
lim⁡n→0−(n(n+2)(n+1)2)=0\lim_{n \to 0^-}\left(\frac{n \left(n + 2\right)}{\left(n + 1\right)^{2}}\right) = 0
More at n→0 from the left
lim⁡n→0+(n(n+2)(n+1)2)=0\lim_{n \to 0^+}\left(\frac{n \left(n + 2\right)}{\left(n + 1\right)^{2}}\right) = 0
More at n→0 from the right
lim⁡n→1−(n(n+2)(n+1)2)=34\lim_{n \to 1^-}\left(\frac{n \left(n + 2\right)}{\left(n + 1\right)^{2}}\right) = \frac{3}{4}
More at n→1 from the left
lim⁡n→1+(n(n+2)(n+1)2)=34\lim_{n \to 1^+}\left(\frac{n \left(n + 2\right)}{\left(n + 1\right)^{2}}\right) = \frac{3}{4}
More at n→1 from the right
lim⁡n→−∞(n(n+2)(n+1)2)=1\lim_{n \to -\infty}\left(\frac{n \left(n + 2\right)}{\left(n + 1\right)^{2}}\right) = 1
More at n→-oo
Rapid solution [src] 
1
11
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The graph
Limit of the function n*(2+n)/(1+n)^2
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