Mister Exam
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-
How to use it?
- Graphing y =:
- -x-7/4
- -x^2+3x+2
- 1/4*(x+4)*(x-2)^2
- y=x^4+3^4-4
- Limit of the function:
-
(4+n)/(5+n)
-
Identical expressions
- (four +n)/(five +n)
- (4 plus n) divide by (5 plus n)
- (four plus n) divide by (five plus n)
- 4+n/5+n
- (4+n) divide by (5+n)
-
Similar expressions
- (4+n)/(5-n)
- (4-n)/(5+n)
Graphing y = (4+n)/(5+n)
The solution
You have entered
[src]
4 + n
f(n) = -----
5 + n
$$f{\left(n \right)} = \frac{n + 4}{n + 5}$$
f = (n + 4)/(n + 5)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$n_{1} = -5$$
$$n_{1} = -5$$
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis N at f = 0
so we need to solve the equation:
$$\frac{n + 4}{n + 5} = 0$$
Solve this equation
The points of intersection with the axis N:
Analytical solution
$$n_{1} = -4$$
Numerical solution
$$n_{1} = -4$$
so we need to solve the equation:
$$\frac{n + 4}{n + 5} = 0$$
Solve this equation
The points of intersection with the axis N:
Analytical solution
$$n_{1} = -4$$
Numerical solution
$$n_{1} = -4$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d n} f{\left(n \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d n} f{\left(n \right)} = $$
the first derivative
$$- \frac{n + 4}{\left(n + 5\right)^{2}} + \frac{1}{n + 5} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d n} f{\left(n \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d n} f{\left(n \right)} = $$
the first derivative
$$- \frac{n + 4}{\left(n + 5\right)^{2}} + \frac{1}{n + 5} = 0$$
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
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = $$
the second derivative
$$\frac{2 \left(\frac{n + 4}{n + 5} - 1\right)}{\left(n + 5\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = $$
the second derivative
$$\frac{2 \left(\frac{n + 4}{n + 5} - 1\right)}{\left(n + 5\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$n_{1} = -5$$
$$n_{1} = -5$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at n->+oo and n->-oo
$$\lim_{n \to -\infty}\left(\frac{n + 4}{n + 5}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{n \to \infty}\left(\frac{n + 4}{n + 5}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{n \to -\infty}\left(\frac{n + 4}{n + 5}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{n \to \infty}\left(\frac{n + 4}{n + 5}\right) = 1$$
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 (4 + n)/(5 + n), divided by n at n->+oo and n ->-oo
$$\lim_{n \to -\infty}\left(\frac{n + 4}{n \left(n + 5\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{n \to \infty}\left(\frac{n + 4}{n \left(n + 5\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{n \to -\infty}\left(\frac{n + 4}{n \left(n + 5\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{n \to \infty}\left(\frac{n + 4}{n \left(n + 5\right)}\right) = 0$$
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(-n) и f = -f(-n).
So, check:
$$\frac{n + 4}{n + 5} = \frac{4 - n}{5 - n}$$
- No
$$\frac{n + 4}{n + 5} = - \frac{4 - n}{5 - n}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{n + 4}{n + 5} = \frac{4 - n}{5 - n}$$
- No
$$\frac{n + 4}{n + 5} = - \frac{4 - n}{5 - n}$$
- No
so, the function
not is
neither even, nor odd