Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (sin(3*x)^2-sin(x)^2)/x^2
-
Limit of ((5+n^2-6*n)/(5+n^2-5*n))^(2+3*n)
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Limit of ((1+n)/n)^(n^2)
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Limit of (n/(1+2*n))^n*((1+n)/(3+2*n))^(-1-n)
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Identical expressions
- ((one +n)/n)^(n^ two)
- ((1 plus n) divide by n) to the power of (n squared )
- ((one plus n) divide by n) to the power of (n to the power of two)
- ((1+n)/n)(n2)
- 1+n/nn2
- ((1+n)/n)^(n²)
- ((1+n)/n) to the power of (n to the power of 2)
- 1+n/n^n^2
- ((1+n) divide by n)^(n^2)
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Similar expressions
- ((1-n)/n)^(n^2)
Limit of the function ((1+n)/n)^(n^2)
The solution
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/ 2\
\n /
/1 + n\
lim |-----|
n->oo\ n /
$$\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n^{2}}$$
Limit(((1 + n)/n)^(n^2), n, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n^{2}}$$
transform
$$\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n^{2}}$$
=
$$\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n^{2}}$$
=
$$\lim_{n \to \infty} \left(\frac{n}{n} + \frac{1}{n}\right)^{n^{2}}$$
=
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n^{2}}$$
=
do replacement
$$u = \frac{n}{1}$$
then
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n^{2}}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u^{2}}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u^{2}}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{u}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{u} = e^{u}$$
The final answer:
$$\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n^{2}} = \infty$$
$$\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n^{2}}$$
transform
$$\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n^{2}}$$
=
$$\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n^{2}}$$
=
$$\lim_{n \to \infty} \left(\frac{n}{n} + \frac{1}{n}\right)^{n^{2}}$$
=
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n^{2}}$$
=
do replacement
$$u = \frac{n}{1}$$
then
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n^{2}}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u^{2}}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u^{2}}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{u}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{u} = e^{u}$$
The final answer:
$$\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n^{2}} = \infty$$
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
$$\lim_{n \to \infty} \left(\frac{n + 1}{n}\right)^{n^{2}} = \infty$$
$$\lim_{n \to 0^-} \left(\frac{n + 1}{n}\right)^{n^{2}} = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+} \left(\frac{n + 1}{n}\right)^{n^{2}} = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-} \left(\frac{n + 1}{n}\right)^{n^{2}} = 2$$
More at n→1 from the left
$$\lim_{n \to 1^+} \left(\frac{n + 1}{n}\right)^{n^{2}} = 2$$
More at n→1 from the right
$$\lim_{n \to -\infty} \left(\frac{n + 1}{n}\right)^{n^{2}} = 0$$
More at n→-oo
$$\lim_{n \to 0^-} \left(\frac{n + 1}{n}\right)^{n^{2}} = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+} \left(\frac{n + 1}{n}\right)^{n^{2}} = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-} \left(\frac{n + 1}{n}\right)^{n^{2}} = 2$$
More at n→1 from the left
$$\lim_{n \to 1^+} \left(\frac{n + 1}{n}\right)^{n^{2}} = 2$$
More at n→1 from the right
$$\lim_{n \to -\infty} \left(\frac{n + 1}{n}\right)^{n^{2}} = 0$$
More at n→-oo
The graph