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

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

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     /        2\
     | (1 + n) |
 lim |---------|
n->oo\n*(2 + n)/
lim⁡n→∞((n+1)2n(n+2))\lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2}}{n \left(n + 2\right)}\right) 
Limit((1 + n)^2/((n*(2 + n))), n, oo, dir='-')
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡n→∞((n+1)2n)=∞\lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2}}{n}\right) = \infty
and limit for the denominator is
lim⁡n→∞(n+2)=∞\lim_{n \to \infty}\left(n + 2\right) = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡n→∞((n+1)2n(n+2))\lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2}}{n \left(n + 2\right)}\right)
=
Let's transform the function under the limit a few
lim⁡n→∞((n+1)2n(n+2))\lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2}}{n \left(n + 2\right)}\right)
=
lim⁡n→∞(ddn(n+1)2nddn(n+2))\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{\left(n + 1\right)^{2}}{n}}{\frac{d}{d n} \left(n + 2\right)}\right)
=
lim⁡n→∞(1−1n2)\lim_{n \to \infty}\left(1 - \frac{1}{n^{2}}\right)
=
lim⁡n→∞(1−1n2)\lim_{n \to \infty}\left(1 - \frac{1}{n^{2}}\right)
=
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
02468-8-6-4-2-1010-1010
Plot the graph
Other limits n→0, -oo, +oo, 1
lim⁡n→∞((n+1)2n(n+2))=1\lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2}}{n \left(n + 2\right)}\right) = 1
lim⁡n→0−((n+1)2n(n+2))=−∞\lim_{n \to 0^-}\left(\frac{\left(n + 1\right)^{2}}{n \left(n + 2\right)}\right) = -\infty
More at n→0 from the left
lim⁡n→0+((n+1)2n(n+2))=∞\lim_{n \to 0^+}\left(\frac{\left(n + 1\right)^{2}}{n \left(n + 2\right)}\right) = \infty
More at n→0 from the right
lim⁡n→1−((n+1)2n(n+2))=43\lim_{n \to 1^-}\left(\frac{\left(n + 1\right)^{2}}{n \left(n + 2\right)}\right) = \frac{4}{3}
More at n→1 from the left
lim⁡n→1+((n+1)2n(n+2))=43\lim_{n \to 1^+}\left(\frac{\left(n + 1\right)^{2}}{n \left(n + 2\right)}\right) = \frac{4}{3}
More at n→1 from the right
lim⁡n→−∞((n+1)2n(n+2))=1\lim_{n \to -\infty}\left(\frac{\left(n + 1\right)^{2}}{n \left(n + 2\right)}\right) = 1
More at n→-oo
Rapid solution [src] 
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11
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The graph
Limit of the function (1+n)^2/(n*(2+n))
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