Mister Exam
Graphing y = (((n+2)/(n+1))^(n+1))^(n+1)/(((n+1)/n)^n)^n
The solution
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$$f{\left(n \right)} = \frac{\left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{n}}$$
f = (((n + 2)/(n + 1))^(n + 1))^(n + 1)/(((n + 1)/n)^n)^n
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$n_{1} = -1$$
$$n_{2} = 0$$
$$n_{1} = -1$$
$$n_{2} = 0$$
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:
$$\frac{\left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{n}} = 0$$
Solve this equation
The points of intersection with the axis N:
Numerical solution
$$n_{1} = -2.87905574633645 \cdot 10^{27}$$
$$n_{2} = 2.44032585178171 \cdot 10^{27}$$
$$n_{3} = -2.55044188820325 \cdot 10^{27}$$
$$n_{4} = -1.61681706542867 \cdot 10^{27}$$
$$n_{5} = -3.68777749909299 \cdot 10^{27}$$
$$n_{6} = -4.70641685979808 \cdot 10^{27}$$
$$n_{7} = -4.16721125437428 \cdot 10^{27}$$
$$n_{8} = 2.39337583823029 \cdot 10^{27}$$
$$n_{9} = -1.5047720970826 \cdot 10^{27}$$
$$n_{10} = 2.34375326698733 \cdot 10^{27}$$
$$n_{11} = -1.60277685318126 \cdot 10^{27}$$
$$n_{12} = -2.65795079248419 \cdot 10^{27}$$
$$n_{13} = -4.1538536533901 \cdot 10^{27}$$
$$n_{14} = 2.38503905287327 \cdot 10^{27}$$
$$n_{15} = -1.42660120446531 \cdot 10^{27}$$
$$n_{16} = -4.44613753075722 \cdot 10^{27}$$
$$n_{17} = -1.31555925446613 \cdot 10^{27}$$
$$n_{18} = -2.31522019436084 \cdot 10^{27}$$
$$n_{19} = -2.29445926639657 \cdot 10^{27}$$
$$n_{20} = -1.66576111206606 \cdot 10^{27}$$
$$n_{21} = -3.87343812910337 \cdot 10^{27}$$
$$n_{22} = -1.73500392853528 \cdot 10^{27}$$
$$n_{23} = 1.52018063217681 \cdot 10^{27}$$
$$n_{24} = -4.03557457157977 \cdot 10^{27}$$
$$n_{25} = -1.58740209950405 \cdot 10^{27}$$
$$n_{26} = -4.43000887536658 \cdot 10^{27}$$
$$n_{27} = -4.212187995345 \cdot 10^{27}$$
$$n_{28} = -1.56141816453821 \cdot 10^{27}$$
$$n_{29} = -1.35920645420237 \cdot 10^{27}$$
$$n_{30} = -3.38206995719475 \cdot 10^{27}$$
so we need to solve the equation:
$$\frac{\left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{n}} = 0$$
Solve this equation
The points of intersection with the axis N:
Numerical solution
$$n_{1} = -2.87905574633645 \cdot 10^{27}$$
$$n_{2} = 2.44032585178171 \cdot 10^{27}$$
$$n_{3} = -2.55044188820325 \cdot 10^{27}$$
$$n_{4} = -1.61681706542867 \cdot 10^{27}$$
$$n_{5} = -3.68777749909299 \cdot 10^{27}$$
$$n_{6} = -4.70641685979808 \cdot 10^{27}$$
$$n_{7} = -4.16721125437428 \cdot 10^{27}$$
$$n_{8} = 2.39337583823029 \cdot 10^{27}$$
$$n_{9} = -1.5047720970826 \cdot 10^{27}$$
$$n_{10} = 2.34375326698733 \cdot 10^{27}$$
$$n_{11} = -1.60277685318126 \cdot 10^{27}$$
$$n_{12} = -2.65795079248419 \cdot 10^{27}$$
$$n_{13} = -4.1538536533901 \cdot 10^{27}$$
$$n_{14} = 2.38503905287327 \cdot 10^{27}$$
$$n_{15} = -1.42660120446531 \cdot 10^{27}$$
$$n_{16} = -4.44613753075722 \cdot 10^{27}$$
$$n_{17} = -1.31555925446613 \cdot 10^{27}$$
$$n_{18} = -2.31522019436084 \cdot 10^{27}$$
$$n_{19} = -2.29445926639657 \cdot 10^{27}$$
$$n_{20} = -1.66576111206606 \cdot 10^{27}$$
$$n_{21} = -3.87343812910337 \cdot 10^{27}$$
$$n_{22} = -1.73500392853528 \cdot 10^{27}$$
$$n_{23} = 1.52018063217681 \cdot 10^{27}$$
$$n_{24} = -4.03557457157977 \cdot 10^{27}$$
$$n_{25} = -1.58740209950405 \cdot 10^{27}$$
$$n_{26} = -4.43000887536658 \cdot 10^{27}$$
$$n_{27} = -4.212187995345 \cdot 10^{27}$$
$$n_{28} = -1.56141816453821 \cdot 10^{27}$$
$$n_{29} = -1.35920645420237 \cdot 10^{27}$$
$$n_{30} = -3.38206995719475 \cdot 10^{27}$$
Vertical asymptotes
Have:
$$n_{1} = -1$$
$$n_{2} = 0$$
$$n_{1} = -1$$
$$n_{2} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at n->+oo and n->-oo
$$\lim_{n \to -\infty}\left(\frac{\left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{n}}\right) = e$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = e$$
$$\lim_{n \to \infty}\left(\frac{\left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{n}}\right) = e$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = e$$
$$\lim_{n \to -\infty}\left(\frac{\left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{n}}\right) = e$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = e$$
$$\lim_{n \to \infty}\left(\frac{\left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{n}}\right) = e$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = e$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (((n + 2)/(n + 1))^(n + 1))^(n + 1)/(((n + 1)/n)^n)^n, divided by n at n->+oo and n ->-oo
$$\lim_{n \to -\infty}\left(\frac{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{- n} \left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{n}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{n \to \infty}\left(\frac{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{- n} \left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{n}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{n \to -\infty}\left(\frac{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{- n} \left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{n}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{n \to \infty}\left(\frac{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{- n} \left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{n}\right) = 0$$
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:
$$\frac{\left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{n}} = \left(\left(- \frac{1 - n}{n}\right)^{- n}\right)^{n} \left(\left(\frac{2 - n}{1 - n}\right)^{1 - n}\right)^{1 - n}$$
- No
$$\frac{\left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{n}} = - \left(\left(- \frac{1 - n}{n}\right)^{- n}\right)^{n} \left(\left(\frac{2 - n}{1 - n}\right)^{1 - n}\right)^{1 - n}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{n}} = \left(\left(- \frac{1 - n}{n}\right)^{- n}\right)^{n} \left(\left(\frac{2 - n}{1 - n}\right)^{1 - n}\right)^{1 - n}$$
- No
$$\frac{\left(\left(\frac{n + 2}{n + 1}\right)^{n + 1}\right)^{n + 1}}{\left(\left(\frac{n + 1}{n}\right)^{n}\right)^{n}} = - \left(\left(- \frac{1 - n}{n}\right)^{- n}\right)^{n} \left(\left(\frac{2 - n}{1 - n}\right)^{1 - n}\right)^{1 - n}$$
- No
so, the function
not is
neither even, nor odd