Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-1+3*sqrt(x))/(-1+4*sqrt(x))
-
Limit of (-4+2*x)/(6+4*x+7*x^2)
-
Limit of ((5+2*x)/(2*x))^(3*x)
-
Limit of ((-4+2*x)/(1+2*x))^(5+3*x)
- Sum of series:
-
(1+n)/(1+n^2)
-
Identical expressions
- (one +n)/(one +n^ two)
- (1 plus n) divide by (1 plus n squared )
- (one plus n) divide by (one plus n to the power of two)
- (1+n)/(1+n2)
- 1+n/1+n2
- (1+n)/(1+n²)
- (1+n)/(1+n to the power of 2)
- 1+n/1+n^2
- (1+n) divide by (1+n^2)
-
Similar expressions
- (1+n)/(1-n^2)
- (1-n)/(1+n^2)
Limit of the function (1+n)/(1+n^2)
The solution
You have entered
[src]
/1 + n \
lim |------|
n->oo| 2|
\1 + n /
$$\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right)$$
Limit((1 + n)/(1 + n^2), n, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right)$$
Let's divide numerator and denominator by n^2:
$$\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right)$$ =
$$\lim_{n \to \infty}\left(\frac{\frac{1}{n} + \frac{1}{n^{2}}}{1 + \frac{1}{n^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{n}$$
then
$$\lim_{n \to \infty}\left(\frac{\frac{1}{n} + \frac{1}{n^{2}}}{1 + \frac{1}{n^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u^{2} + u}{u^{2} + 1}\right)$$
=
$$\frac{0^{2}}{0^{2} + 1} = 0$$
The final answer:
$$\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right) = 0$$
$$\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right)$$
Let's divide numerator and denominator by n^2:
$$\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right)$$ =
$$\lim_{n \to \infty}\left(\frac{\frac{1}{n} + \frac{1}{n^{2}}}{1 + \frac{1}{n^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{n}$$
then
$$\lim_{n \to \infty}\left(\frac{\frac{1}{n} + \frac{1}{n^{2}}}{1 + \frac{1}{n^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u^{2} + u}{u^{2} + 1}\right)$$
=
$$\frac{0^{2}}{0^{2} + 1} = 0$$
The final answer:
$$\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right) = 0$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{n \to \infty}\left(n + 1\right) = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty}\left(n^{2} + 1\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(n + 1\right)}{\frac{d}{d n} \left(n^{2} + 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{1}{2 n}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{1}{2 n}\right)$$
=
$$0$$
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)
oo/oo,
i.e. limit for the numerator is
$$\lim_{n \to \infty}\left(n + 1\right) = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty}\left(n^{2} + 1\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(n + 1\right)}{\frac{d}{d n} \left(n^{2} + 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{1}{2 n}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{1}{2 n}\right)$$
=
$$0$$
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
Other limits n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{n + 1}{n^{2} + 1}\right) = 0$$
$$\lim_{n \to 0^-}\left(\frac{n + 1}{n^{2} + 1}\right) = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{n + 1}{n^{2} + 1}\right) = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{n + 1}{n^{2} + 1}\right) = 1$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{n + 1}{n^{2} + 1}\right) = 1$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{n + 1}{n^{2} + 1}\right) = 0$$
More at n→-oo
$$\lim_{n \to 0^-}\left(\frac{n + 1}{n^{2} + 1}\right) = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{n + 1}{n^{2} + 1}\right) = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{n + 1}{n^{2} + 1}\right) = 1$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{n + 1}{n^{2} + 1}\right) = 1$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{n + 1}{n^{2} + 1}\right) = 0$$
More at n→-oo
The graph