Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-2*x^2+4*x^3+5*x)/(3*x^2+7*x)
-
Limit of (-cos(5*x)+cos(3*x))/x^2
-
Limit of (-1+cos(7*x))/(-1+cos(3*x))
-
Limit of (16+x^2+10*x)/(-6+x^2-x)
- Sum of series:
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1/(1+n^2)
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Identical expressions
- one /(one +n^ two)
- 1 divide by (1 plus n squared )
- one divide by (one plus n to the power of two)
- 1/(1+n2)
- 1/1+n2
- 1/(1+n²)
- 1/(1+n to the power of 2)
- 1/1+n^2
- 1 divide by (1+n^2)
-
Similar expressions
- 1/(1-n^2)
Limit of the function 1/(1+n^2)
The solution
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1
lim ------
n->oo 2
1 + n
$$\lim_{n \to \infty} \frac{1}{n^{2} + 1}$$
Limit(1/(1 + n^2), n, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{n \to \infty} \frac{1}{n^{2} + 1}$$
Let's divide numerator and denominator by n^2:
$$\lim_{n \to \infty} \frac{1}{n^{2} + 1}$$ =
$$\lim_{n \to \infty}\left(\frac{1}{n^{2} \left(1 + \frac{1}{n^{2}}\right)}\right)$$
Do Replacement
$$u = \frac{1}{n}$$
then
$$\lim_{n \to \infty}\left(\frac{1}{n^{2} \left(1 + \frac{1}{n^{2}}\right)}\right) = \lim_{u \to 0^+}\left(\frac{u^{2}}{u^{2} + 1}\right)$$
=
$$\frac{0^{2}}{0^{2} + 1} = 0$$
The final answer:
$$\lim_{n \to \infty} \frac{1}{n^{2} + 1} = 0$$
$$\lim_{n \to \infty} \frac{1}{n^{2} + 1}$$
Let's divide numerator and denominator by n^2:
$$\lim_{n \to \infty} \frac{1}{n^{2} + 1}$$ =
$$\lim_{n \to \infty}\left(\frac{1}{n^{2} \left(1 + \frac{1}{n^{2}}\right)}\right)$$
Do Replacement
$$u = \frac{1}{n}$$
then
$$\lim_{n \to \infty}\left(\frac{1}{n^{2} \left(1 + \frac{1}{n^{2}}\right)}\right) = \lim_{u \to 0^+}\left(\frac{u^{2}}{u^{2} + 1}\right)$$
=
$$\frac{0^{2}}{0^{2} + 1} = 0$$
The final answer:
$$\lim_{n \to \infty} \frac{1}{n^{2} + 1} = 0$$
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
Other limits n→0, -oo, +oo, 1
$$\lim_{n \to \infty} \frac{1}{n^{2} + 1} = 0$$
$$\lim_{n \to 0^-} \frac{1}{n^{2} + 1} = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+} \frac{1}{n^{2} + 1} = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-} \frac{1}{n^{2} + 1} = \frac{1}{2}$$
More at n→1 from the left
$$\lim_{n \to 1^+} \frac{1}{n^{2} + 1} = \frac{1}{2}$$
More at n→1 from the right
$$\lim_{n \to -\infty} \frac{1}{n^{2} + 1} = 0$$
More at n→-oo
$$\lim_{n \to 0^-} \frac{1}{n^{2} + 1} = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+} \frac{1}{n^{2} + 1} = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-} \frac{1}{n^{2} + 1} = \frac{1}{2}$$
More at n→1 from the left
$$\lim_{n \to 1^+} \frac{1}{n^{2} + 1} = \frac{1}{2}$$
More at n→1 from the right
$$\lim_{n \to -\infty} \frac{1}{n^{2} + 1} = 0$$
More at n→-oo
The graph