Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
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Limit of (1+2*n)/(-1+3*n)
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Limit of (1+1/n)^(3*n)
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Limit of (6+x^2-5*x)/(2+x^3-x-2*x^2)
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Limit of (-6+x^2+5*x)/(-1+x)
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Identical expressions
- (one + one /n)^(three *n)
- (1 plus 1 divide by n) to the power of (3 multiply by n)
- (one plus one divide by n) to the power of (three multiply by n)
- (1+1/n)(3*n)
- 1+1/n3*n
- (1+1/n)^(3n)
- (1+1/n)(3n)
- 1+1/n3n
- 1+1/n^3n
- (1+1 divide by n)^(3*n)
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Similar expressions
- (1-1/n)^(3*n)
Limit of the function (1+1/n)^(3*n)
The solution
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3*n
/ 1\
lim |1 + -|
n->oo\ n/
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{3 n}$$
Limit((1 + 1/n)^(3*n), n, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{3 n}$$
transform
do replacement
$$u = \frac{n}{1}$$
then
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{3 n}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{3 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{3 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{3}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{3} = e^{3}$$
The final answer:
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{3 n} = e^{3}$$
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{3 n}$$
transform
do replacement
$$u = \frac{n}{1}$$
then
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{3 n}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{3 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{3 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{3}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{3} = e^{3}$$
The final answer:
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{3 n} = e^{3}$$
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
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{3 n} = e^{3}$$
$$\lim_{n \to 0^-} \left(1 + \frac{1}{n}\right)^{3 n} = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+} \left(1 + \frac{1}{n}\right)^{3 n} = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-} \left(1 + \frac{1}{n}\right)^{3 n} = 8$$
More at n→1 from the left
$$\lim_{n \to 1^+} \left(1 + \frac{1}{n}\right)^{3 n} = 8$$
More at n→1 from the right
$$\lim_{n \to -\infty} \left(1 + \frac{1}{n}\right)^{3 n} = e^{3}$$
More at n→-oo
$$\lim_{n \to 0^-} \left(1 + \frac{1}{n}\right)^{3 n} = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+} \left(1 + \frac{1}{n}\right)^{3 n} = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-} \left(1 + \frac{1}{n}\right)^{3 n} = 8$$
More at n→1 from the left
$$\lim_{n \to 1^+} \left(1 + \frac{1}{n}\right)^{3 n} = 8$$
More at n→1 from the right
$$\lim_{n \to -\infty} \left(1 + \frac{1}{n}\right)^{3 n} = e^{3}$$
More at n→-oo
The graph