Mister Exam
Other calculators
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-
How to use it?
- Graphing y =:
- (6-x^3)/(x^2)
- x+cosx
- -x^4+4x^3+1
- x^3+x^2-x-1
- Limit of the function:
-
(-exp(-x)+exp(x))/x
-
Identical expressions
- (-exp(-x)+exp(x))/x
- ( minus exponent of ( minus x) plus exponent of (x)) divide by x
- -exp-x+expx/x
- (-exp(-x)+exp(x)) divide by x
-
Similar expressions
- (-exp(-x)-exp(x))/x
- (-exp(x)+exp(x))/x
- (exp(-x)+exp(x))/x
Graphing y = (-exp(-x)+exp(x))/x
The solution
You have entered
[src]
-x x
- e + e
f(x) = ----------
x
$$f{\left(x \right)} = \frac{e^{x} - e^{- x}}{x}$$
f = (exp(x) - exp(-x))/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:
$$\frac{e^{x} - e^{- x}}{x} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\frac{e^{x} - e^{- x}}{x} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$\frac{e^{x} + e^{- x}}{x} - \frac{e^{x} - e^{- x}}{x^{2}} = 0$$
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
$$\frac{e^{x} + e^{- x}}{x} - \frac{e^{x} - e^{- x}}{x^{2}} = 0$$
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
$$\frac{e^{x} - e^{- x} - \frac{2 \left(e^{x} + e^{- x}\right)}{x} + \frac{2 \left(e^{x} - e^{- x}\right)}{x^{2}}}{x} = 0$$
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
$$\frac{e^{x} - e^{- x} - \frac{2 \left(e^{x} + e^{- x}\right)}{x} + \frac{2 \left(e^{x} - e^{- x}\right)}{x^{2}}}{x} = 0$$
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
$$\lim_{x \to -\infty}\left(\frac{e^{x} - e^{- x}}{x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{e^{x} - e^{- x}}{x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{e^{x} - e^{- x}}{x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{e^{x} - e^{- x}}{x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (-exp(-x) + exp(x))/x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{e^{x} - e^{- x}}{x^{2}}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{e^{x} - e^{- x}}{x^{2}}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{e^{x} - e^{- x}}{x^{2}}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{e^{x} - e^{- x}}{x^{2}}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$\frac{e^{x} - e^{- x}}{x} = - \frac{- e^{x} + e^{- x}}{x}$$
- No
$$\frac{e^{x} - e^{- x}}{x} = \frac{- e^{x} + e^{- x}}{x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{e^{x} - e^{- x}}{x} = - \frac{- e^{x} + e^{- x}}{x}$$
- No
$$\frac{e^{x} - e^{- x}}{x} = \frac{- e^{x} + e^{- x}}{x}$$
- No
so, the function
not is
neither even, nor odd