Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
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Limit of (-exp(-x)+exp(x))/(exp(x)+exp(-x))
-
Limit of (3+3*x^3+5*x+5*x^2)/(-1+x^2)
-
Limit of (3-x^2+5*x)/(4*x^7+81*x)
-
Limit of -cos(1/x)+2*x*sin(1/x)
- Sum of series:
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(1+5/n)^n
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Identical expressions
- (one + five /n)^n
- (1 plus 5 divide by n) to the power of n
- (one plus five divide by n) to the power of n
- (1+5/n)n
- 1+5/nn
- 1+5/n^n
- (1+5 divide by n)^n
-
Similar expressions
- (1-5/n)^n
Limit of the function (1+5/n)^n
The solution
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n
/ 5\
lim |1 + -|
n->oo\ n/
$$\lim_{n \to \infty} \left(1 + \frac{5}{n}\right)^{n}$$
Limit((1 + 5/n)^n, n, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{n \to \infty} \left(1 + \frac{5}{n}\right)^{n}$$
transform
do replacement
$$u = \frac{n}{5}$$
then
$$\lim_{n \to \infty} \left(1 + \frac{5}{n}\right)^{n}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{5 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{5 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{5}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{5} = e^{5}$$
The final answer:
$$\lim_{n \to \infty} \left(1 + \frac{5}{n}\right)^{n} = e^{5}$$
$$\lim_{n \to \infty} \left(1 + \frac{5}{n}\right)^{n}$$
transform
do replacement
$$u = \frac{n}{5}$$
then
$$\lim_{n \to \infty} \left(1 + \frac{5}{n}\right)^{n}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{5 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{5 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{5}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{5} = e^{5}$$
The final answer:
$$\lim_{n \to \infty} \left(1 + \frac{5}{n}\right)^{n} = e^{5}$$
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
$$\lim_{n \to \infty} \left(1 + \frac{5}{n}\right)^{n} = e^{5}$$
$$\lim_{n \to 0^-} \left(1 + \frac{5}{n}\right)^{n} = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+} \left(1 + \frac{5}{n}\right)^{n} = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-} \left(1 + \frac{5}{n}\right)^{n} = 6$$
More at n→1 from the left
$$\lim_{n \to 1^+} \left(1 + \frac{5}{n}\right)^{n} = 6$$
More at n→1 from the right
$$\lim_{n \to -\infty} \left(1 + \frac{5}{n}\right)^{n} = e^{5}$$
More at n→-oo
$$\lim_{n \to 0^-} \left(1 + \frac{5}{n}\right)^{n} = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+} \left(1 + \frac{5}{n}\right)^{n} = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-} \left(1 + \frac{5}{n}\right)^{n} = 6$$
More at n→1 from the left
$$\lim_{n \to 1^+} \left(1 + \frac{5}{n}\right)^{n} = 6$$
More at n→1 from the right
$$\lim_{n \to -\infty} \left(1 + \frac{5}{n}\right)^{n} = e^{5}$$
More at n→-oo
The graph