Mister Exam
Graphing y = exp(x/(x^2-1))
The solution
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x
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2
x - 1
f(x) = e
$$f{\left(x \right)} = e^{\frac{x}{x^{2} - 1}}$$
f = exp(x/(x^2 - 1))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{1} = -1$$
$$x_{2} = 1$$
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:
$$e^{\frac{x}{x^{2} - 1}} = 0$$
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:
$$e^{\frac{x}{x^{2} - 1}} = 0$$
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
$$\left(- \frac{2 x^{2}}{\left(x^{2} - 1\right)^{2}} + \frac{1}{x^{2} - 1}\right) e^{\frac{x}{x^{2} - 1}} = 0$$
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
$$\left(- \frac{2 x^{2}}{\left(x^{2} - 1\right)^{2}} + \frac{1}{x^{2} - 1}\right) e^{\frac{x}{x^{2} - 1}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Vertical asymptotes
Have:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{1} = -1$$
$$x_{2} = 1$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} e^{\frac{x}{x^{2} - 1}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} e^{\frac{x}{x^{2} - 1}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty} e^{\frac{x}{x^{2} - 1}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} e^{\frac{x}{x^{2} - 1}} = 1$$
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 exp(x/(x^2 - 1)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{e^{\frac{x}{x^{2} - 1}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{e^{\frac{x}{x^{2} - 1}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{e^{\frac{x}{x^{2} - 1}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{e^{\frac{x}{x^{2} - 1}}}{x}\right) = 0$$
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:
$$e^{\frac{x}{x^{2} - 1}} = e^{- \frac{x}{x^{2} - 1}}$$
- No
$$e^{\frac{x}{x^{2} - 1}} = - e^{- \frac{x}{x^{2} - 1}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$e^{\frac{x}{x^{2} - 1}} = e^{- \frac{x}{x^{2} - 1}}$$
- No
$$e^{\frac{x}{x^{2} - 1}} = - e^{- \frac{x}{x^{2} - 1}}$$
- No
so, the function
not is
neither even, nor odd