Mister Exam
Other calculators
- Graph of implicit function
- Derivative Step by Step
- Integral Step by Step
- Graph of parametric function
- Graph in Polar Coordinates
- Surface defined parametrically
- Surface defined by equation
- Curve in 3D space defined parametrically
- Area between lines
- The intersection points of functions
-
How to use it?
- Graphing y =:
- (x-5)/(x-3)
- -x+5
- x^4-4x^3+4
- x^4-2x^3+3
- Limit of the function:
-
(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)
Graphing y = (1-x)^(1/x)
The solution
You have entered
[src]
x _______ f(x) = \/ 1 - x
$$f{\left(x \right)} = \left(1 - x\right)^{\frac{1}{x}}$$
f = (1 - x)^(1/x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\left(1 - x\right)^{\frac{1}{x}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
so we need to solve the equation:
$$\left(1 - x\right)^{\frac{1}{x}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\left(1 - x\right)^{\frac{1}{x}} \left(- \frac{1}{x \left(1 - x\right)} - \frac{\log{\left(1 - x \right)}}{x^{2}}\right) = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\left(1 - x\right)^{\frac{1}{x}} \left(- \frac{1}{x \left(1 - x\right)} - \frac{\log{\left(1 - x \right)}}{x^{2}}\right) = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\left(1 - x\right)^{\frac{1}{x}} \left(- \frac{1}{\left(x - 1\right)^{2}} + \frac{\left(\frac{1}{x - 1} - \frac{\log{\left(1 - x \right)}}{x}\right)^{2}}{x} - \frac{2}{x \left(x - 1\right)} + \frac{2 \log{\left(1 - x \right)}}{x^{2}}\right)}{x} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\left(1 - x\right)^{\frac{1}{x}} \left(- \frac{1}{\left(x - 1\right)^{2}} + \frac{\left(\frac{1}{x - 1} - \frac{\log{\left(1 - x \right)}}{x}\right)^{2}}{x} - \frac{2}{x \left(x - 1\right)} + \frac{2 \log{\left(1 - x \right)}}{x^{2}}\right)}{x} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \left(1 - x\right)^{\frac{1}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \left(1 - x\right)^{\frac{1}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty} \left(1 - x\right)^{\frac{1}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} \left(1 - x\right)^{\frac{1}{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (1 - x)^(1/x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(1 - x\right)^{\frac{1}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(1 - x\right)^{\frac{1}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left(1 - x\right)^{\frac{1}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(1 - x\right)^{\frac{1}{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$\left(1 - x\right)^{\frac{1}{x}} = \left(x + 1\right)^{- \frac{1}{x}}$$
- No
$$\left(1 - x\right)^{\frac{1}{x}} = - \left(x + 1\right)^{- \frac{1}{x}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(1 - x\right)^{\frac{1}{x}} = \left(x + 1\right)^{- \frac{1}{x}}$$
- No
$$\left(1 - x\right)^{\frac{1}{x}} = - \left(x + 1\right)^{- \frac{1}{x}}$$
- No
so, the function
not is
neither even, nor odd