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

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

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     /1 + n \
 lim |------|
n->oo|     2|
     \1 + n /
lim⁡n→∞(n+1n2+1)\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right) 
Limit((1 + n)/(1 + n^2), n, oo, dir='-')
Detail solution
Let's take the limit
lim⁡n→∞(n+1n2+1)\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right)
Let's divide numerator and denominator by n^2:
lim⁡n→∞(n+1n2+1)\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right) =
lim⁡n→∞(1n+1n21+1n2)\lim_{n \to \infty}\left(\frac{\frac{1}{n} + \frac{1}{n^{2}}}{1 + \frac{1}{n^{2}}}\right)
Do Replacement
u=1nu = \frac{1}{n}
then
lim⁡n→∞(1n+1n21+1n2)=lim⁡u→0+(u2+uu2+1)\lim_{n \to \infty}\left(\frac{\frac{1}{n} + \frac{1}{n^{2}}}{1 + \frac{1}{n^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u^{2} + u}{u^{2} + 1}\right)
=
0202+1=0\frac{0^{2}}{0^{2} + 1} = 0

The final answer:
lim⁡n→∞(n+1n2+1)=0\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right) = 0
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡n→∞(n+1)=∞\lim_{n \to \infty}\left(n + 1\right) = \infty
and limit for the denominator is
lim⁡n→∞(n2+1)=∞\lim_{n \to \infty}\left(n^{2} + 1\right) = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡n→∞(n+1n2+1)\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right)
=
lim⁡n→∞(ddn(n+1)ddn(n2+1))\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(n + 1\right)}{\frac{d}{d n} \left(n^{2} + 1\right)}\right)
=
lim⁡n→∞(12n)\lim_{n \to \infty}\left(\frac{1}{2 n}\right)
=
lim⁡n→∞(12n)\lim_{n \to \infty}\left(\frac{1}{2 n}\right)
=
00
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
02468-8-6-4-2-10102-2
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Rapid solution [src] 
0
00
Expand and simplify
Other limits n→0, -oo, +oo, 1
lim⁡n→∞(n+1n2+1)=0\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right) = 0
lim⁡n→0−(n+1n2+1)=1\lim_{n \to 0^-}\left(\frac{n + 1}{n^{2} + 1}\right) = 1
More at n→0 from the left
lim⁡n→0+(n+1n2+1)=1\lim_{n \to 0^+}\left(\frac{n + 1}{n^{2} + 1}\right) = 1
More at n→0 from the right
lim⁡n→1−(n+1n2+1)=1\lim_{n \to 1^-}\left(\frac{n + 1}{n^{2} + 1}\right) = 1
More at n→1 from the left
lim⁡n→1+(n+1n2+1)=1\lim_{n \to 1^+}\left(\frac{n + 1}{n^{2} + 1}\right) = 1
More at n→1 from the right
lim⁡n→−∞(n+1n2+1)=0\lim_{n \to -\infty}\left(\frac{n + 1}{n^{2} + 1}\right) = 0
More at n→-oo
The graph
Limit of the function (1+n)/(1+n^2)
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