Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-1+4*n)/(1+2*n)
-
Limit of (4^n-9^n)/(4^(1+n)+9^(1+n))
-
Limit of (1+1/x)^(-1+x)
-
Limit of 7+x^2-14*x/3
-
Identical expressions
- (one + one /x)^(- one +x)
- (1 plus 1 divide by x) to the power of ( minus 1 plus x)
- (one plus one divide by x) to the power of ( minus one plus x)
- (1+1/x)(-1+x)
- 1+1/x-1+x
- 1+1/x^-1+x
- (1+1 divide by x)^(-1+x)
-
Similar expressions
- (1-1/x)^(-1+x)
- (1+1/x)^(1+x)
- (1+1/x)^(-1-x)
Limit of the function (1+1/x)^(-1+x)
The solution
You have entered
[src]
-1 + x
/ 1\
lim |1 + -|
x->oo\ x/
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x - 1}$$
Limit((1 + 1/x)^(-1 + x), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x - 1}$$
transform
do replacement
$$u = \frac{x}{1}$$
then
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x - 1}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u - 1}$$
=
$$\lim_{u \to \infty}\left(\frac{\left(1 + \frac{1}{u}\right)^{u}}{1 + \frac{1}{u}}\right)$$
=
$$\lim_{u \to \infty} \frac{1}{1 + \frac{1}{u}} \lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)$$
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:
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x - 1} = e$$
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x - 1}$$
transform
do replacement
$$u = \frac{x}{1}$$
then
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x - 1}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u - 1}$$
=
$$\lim_{u \to \infty}\left(\frac{\left(1 + \frac{1}{u}\right)^{u}}{1 + \frac{1}{u}}\right)$$
=
$$\lim_{u \to \infty} \frac{1}{1 + \frac{1}{u}} \lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)$$
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:
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x - 1} = e$$
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 x→0, -oo, +oo, 1
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x - 1} = e$$
$$\lim_{x \to 0^-} \left(1 + \frac{1}{x}\right)^{x - 1} = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 + \frac{1}{x}\right)^{x - 1} = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(1 + \frac{1}{x}\right)^{x - 1} = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 + \frac{1}{x}\right)^{x - 1} = 1$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 + \frac{1}{x}\right)^{x - 1} = e$$
More at x→-oo
$$\lim_{x \to 0^-} \left(1 + \frac{1}{x}\right)^{x - 1} = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 + \frac{1}{x}\right)^{x - 1} = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(1 + \frac{1}{x}\right)^{x - 1} = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 + \frac{1}{x}\right)^{x - 1} = 1$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 + \frac{1}{x}\right)^{x - 1} = e$$
More at x→-oo
The graph