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