Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (9+3*x^2+4*x)/(7-7*x+3*x^2)
-
Limit of -pi+(1+cos(x))/x
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Limit of (1-5*x)^(1/x)
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Limit of (1+2/n)^(4*n)
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Identical expressions
- (one + two /n)^(four *n)
- (1 plus 2 divide by n) to the power of (4 multiply by n)
- (one plus two divide by n) to the power of (four multiply by n)
- (1+2/n)(4*n)
- 1+2/n4*n
- (1+2/n)^(4n)
- (1+2/n)(4n)
- 1+2/n4n
- 1+2/n^4n
- (1+2 divide by n)^(4*n)
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Similar expressions
- (1-2/n)^(4*n)
Limit of the function (1+2/n)^(4*n)
The solution
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4*n
/ 2\
lim |1 + -|
n->oo\ n/
$$\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^{4 n}$$
Limit((1 + 2/n)^(4*n), n, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^{4 n}$$
transform
do replacement
$$u = \frac{n}{2}$$
then
$$\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^{4 n}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{8 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{8 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{8}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{8} = e^{8}$$
The final answer:
$$\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^{4 n} = e^{8}$$
$$\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^{4 n}$$
transform
do replacement
$$u = \frac{n}{2}$$
then
$$\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^{4 n}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{8 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{8 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{8}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{8} = e^{8}$$
The final answer:
$$\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^{4 n} = e^{8}$$
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
$$\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^{4 n} = e^{8}$$
$$\lim_{n \to 0^-} \left(1 + \frac{2}{n}\right)^{4 n} = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+} \left(1 + \frac{2}{n}\right)^{4 n} = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-} \left(1 + \frac{2}{n}\right)^{4 n} = 81$$
More at n→1 from the left
$$\lim_{n \to 1^+} \left(1 + \frac{2}{n}\right)^{4 n} = 81$$
More at n→1 from the right
$$\lim_{n \to -\infty} \left(1 + \frac{2}{n}\right)^{4 n} = e^{8}$$
More at n→-oo
$$\lim_{n \to 0^-} \left(1 + \frac{2}{n}\right)^{4 n} = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+} \left(1 + \frac{2}{n}\right)^{4 n} = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-} \left(1 + \frac{2}{n}\right)^{4 n} = 81$$
More at n→1 from the left
$$\lim_{n \to 1^+} \left(1 + \frac{2}{n}\right)^{4 n} = 81$$
More at n→1 from the right
$$\lim_{n \to -\infty} \left(1 + \frac{2}{n}\right)^{4 n} = e^{8}$$
More at n→-oo
The graph