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(n+1)/n
Plotting a function graph/ (n+1)/n

Graphing y = (n+1)/n

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       n + 1
f(n) = -----
         n
f(n)=n+1nf{\left(n \right)} = \frac{n + 1}{n} 
f = (n + 1)/n
The graph of the function
02468-8-6-4-2-1010-5050
The domain of the function
The points at which the function is not precisely defined:
n1=0n_{1} = 0
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:
n+1n=0\frac{n + 1}{n} = 0
Solve this equation
The points of intersection with the axis N:

Analytical solution
n1=−1n_{1} = -1
Numerical solution
n1=−1n_{1} = -1
The points of intersection with the Y axis coordinate
The graph crosses Y axis when n equals 0:
substitute n = 0 to (n + 1)/n.
0+10\frac{0 + 1}{0}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
Extrema of the function
In order to find the extrema, we need to solve the equation
ddnf(n)=0\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:
ddnf(n)=\frac{d}{d n} f{\left(n \right)} =
the first derivative
1n−n+1n2=0\frac{1}{n} - \frac{n + 1}{n^{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
d2dn2f(n)=0\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:
d2dn2f(n)=\frac{d^{2}}{d n^{2}} f{\left(n \right)} =
the second derivative
2(−1+n+1n)n2=0\frac{2 \left(-1 + \frac{n + 1}{n}\right)}{n^{2}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
n1=0n_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at n->+oo and n->-oo
lim⁡n→−∞(n+1n)=1\lim_{n \to -\infty}\left(\frac{n + 1}{n}\right) = 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=1y = 1
lim⁡n→∞(n+1n)=1\lim_{n \to \infty}\left(\frac{n + 1}{n}\right) = 1
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=1y = 1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (n + 1)/n, divided by n at n->+oo and n ->-oo
lim⁡n→−∞(n+1n2)=0\lim_{n \to -\infty}\left(\frac{n + 1}{n^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡n→∞(n+1n2)=0\lim_{n \to \infty}\left(\frac{n + 1}{n^{2}}\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:
n+1n=−−n+1n\frac{n + 1}{n} = - \frac{- n + 1}{n}
- No
n+1n=−n+1n\frac{n + 1}{n} = \frac{- n + 1}{n}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = (n+1)/n
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