Leave the whole number alone for now. Divide 41 by 80 and you get 0.5125
So your answer is 9.5125
For this case what we must do is find a quadratic function that is already factored.
This is because in the factored quadratic equations, it is easier to observe the zeros of the function.
In this case, the zeros of the function represent the time at which the company did not make any profit.
We have the following equation:
p (t) = 40 (t - 3) (t + 2) (t - 5) (t + 3)
We observed that there was no gain in:
t = 3
t = 5
The other roots are discarded because they are negative
Answer:
a.p (t) = 40 (t - 3) (t + 2) (t - 5) (t + 3)
You could graph this, but it is simpler in this case to do it mathematically.
Since the x and values are the same (implying undefined slope) you can subtract the smaller y value from the larger y value. 2.5-1.2=1.3.
Step-by-step explanation:

Given expression is

To, evaluate this limit, let we simplify numerator and denominator individually.
So, Consider Numerator

Clearly, if forms a Geometric progression with first term n and common ratio n respectively.
So, using Sum of n terms of GP, we get


Now, Consider Denominator, we have

can be rewritten as

![\rm \: = \: {n}^{n}\bigg[1 +\bigg[{\dfrac{n - 1}{n}\bigg]}^{n} + \bigg[{\dfrac{n - 2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%20%7Bn%7D%5E%7Bn%7D%5Cbigg%5B1%20%2B%5Cbigg%5B%7B%5Cdfrac%7Bn%20-%201%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7Bn%20-%202%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D)
![\rm \: = \: {n}^{n}\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%20%7Bn%7D%5E%7Bn%7D%5Cbigg%5B1%20%2B%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B2%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D)
Now, Consider

So, on substituting the values evaluated above, we get
![\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{\dfrac{ {n}^{n} - 1}{1 - \dfrac{1}{n} }}{{n}^{n}\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Cfrac%7B%5Cdfrac%7B%20%7Bn%7D%5E%7Bn%7D%20%20-%201%7D%7B1%20-%20%20%5Cdfrac%7B1%7D%7Bn%7D%20%7D%7D%7B%7Bn%7D%5E%7Bn%7D%5Cbigg%5B1%20%2B%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B2%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D%7D%20)
![\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{ {n}^{n} - 1}{{n}^{n}\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Cfrac%7B%20%7Bn%7D%5E%7Bn%7D%20%20-%201%7D%7B%7Bn%7D%5E%7Bn%7D%5Cbigg%5B1%20%2B%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B2%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D%7D%20)
![\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{ {n}^{n}\bigg[1 - \dfrac{1}{ {n}^{n} } \bigg]}{{n}^{n}\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Cfrac%7B%20%7Bn%7D%5E%7Bn%7D%5Cbigg%5B1%20-%20%5Cdfrac%7B1%7D%7B%20%7Bn%7D%5E%7Bn%7D%20%7D%20%5Cbigg%5D%7D%7B%7Bn%7D%5E%7Bn%7D%5Cbigg%5B1%20%2B%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B2%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D%7D%20)
![\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{\bigg[1 - \dfrac{1}{ {n}^{n} } \bigg]}{\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Cfrac%7B%5Cbigg%5B1%20-%20%5Cdfrac%7B1%7D%7B%20%7Bn%7D%5E%7Bn%7D%20%7D%20%5Cbigg%5D%7D%7B%5Cbigg%5B1%20%2B%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B2%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D%7D%20)
![\rm \: = \: \displaystyle\lim_{n \to \infty} \frac{1}{\bigg[1 +\bigg[1 - {\dfrac{1}{n}\bigg]}^{n} + \bigg[1 - {\dfrac{2}{n}\bigg]}^{n} + - - - + \bigg[{\dfrac{1}{n}\bigg]}^{n} \bigg]}](https://tex.z-dn.net/?f=%5Crm%20%5C%3A%20%20%3D%20%20%5C%3A%20%5Cdisplaystyle%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Cfrac%7B1%7D%7B%5Cbigg%5B1%20%2B%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%5Cbigg%5B1%20-%20%7B%5Cdfrac%7B2%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%2B%20%20-%20%20-%20%20-%20%20%2B%20%5Cbigg%5B%7B%5Cdfrac%7B1%7D%7Bn%7D%5Cbigg%5D%7D%5E%7Bn%7D%20%5Cbigg%5D%7D%20)
Now, we know that,
![\red{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\lim_{x \to \infty} \bigg[1 + \dfrac{k}{x} \bigg]^{x} = {e}^{k}}}}](https://tex.z-dn.net/?f=%5Cred%7B%5Crm%20%3A%5Clongmapsto%5C%3A%5Cboxed%7B%5Ctt%7B%20%5Cdisplaystyle%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cbigg%5B1%20%2B%20%5Cdfrac%7Bk%7D%7Bx%7D%20%5Cbigg%5D%5E%7Bx%7D%20%20%3D%20%20%7Be%7D%5E%7Bk%7D%7D%7D%7D%20)
So, using this, we get

Now, in denominator, its an infinite GP series with common ratio 1/e ( < 1 ) and first term 1, so using sum to infinite GP series, we have





Hence,
