Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (1-x)^(1/x)
- Limit of (e^(a*x)-e^(b*x))/sin(x)
-
Limit of ((-1+5*x^2)/(-1+7*x^2))^(1+5*x)
-
Limit of 5*x^4
- Graphing y =:
- (1-x)^(1/x)
-
Identical expressions
- (one -x)^(one /x)
- (1 minus x) to the power of (1 divide by x)
- (one minus x) to the power of (one divide by x)
- (1-x)(1/x)
- 1-x1/x
- 1-x^1/x
- (1-x)^(1 divide by x)
-
Similar expressions
- (1+x)^(1/x)
Limit of the function (1-x)^(1/x)
The solution
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[src]
x _______ lim \/ 1 - x x->0+
$$\lim_{x \to 0^+} \left(1 - x\right)^{\frac{1}{x}}$$
Limit((1 - x)^(1/x), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+} \left(1 - x\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{\left(-1\right) x}$$
then
$$\lim_{x \to 0^+} \left(1 - \frac{1}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-1}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-1} = e^{-1}$$
The final answer:
$$\lim_{x \to 0^+} \left(1 - x\right)^{\frac{1}{x}} = e^{-1}$$
$$\lim_{x \to 0^+} \left(1 - x\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{\left(-1\right) x}$$
then
$$\lim_{x \to 0^+} \left(1 - \frac{1}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-1}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-1} = e^{-1}$$
The final answer:
$$\lim_{x \to 0^+} \left(1 - x\right)^{\frac{1}{x}} = e^{-1}$$
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
One‐sided limits
[src]
x _______ lim \/ 1 - x x->0+
$$\lim_{x \to 0^+} \left(1 - x\right)^{\frac{1}{x}}$$
-1 e
$$e^{-1}$$
= 0.367879441171442
x _______ lim \/ 1 - x x->0-
$$\lim_{x \to 0^-} \left(1 - x\right)^{\frac{1}{x}}$$
-1 e
$$e^{-1}$$
= 0.367879441171442
= 0.367879441171442
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-} \left(1 - x\right)^{\frac{1}{x}} = e^{-1}$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 - x\right)^{\frac{1}{x}} = e^{-1}$$
$$\lim_{x \to \infty} \left(1 - x\right)^{\frac{1}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(1 - x\right)^{\frac{1}{x}} = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 - x\right)^{\frac{1}{x}} = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 - x\right)^{\frac{1}{x}} = 1$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 - x\right)^{\frac{1}{x}} = e^{-1}$$
$$\lim_{x \to \infty} \left(1 - x\right)^{\frac{1}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(1 - x\right)^{\frac{1}{x}} = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 - x\right)^{\frac{1}{x}} = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 - x\right)^{\frac{1}{x}} = 1$$
More at x→-oo
The graph