Answer:
y = ![\frac{4}{3} x](https://tex.z-dn.net/?f=%5Cfrac%7B4%7D%7B3%7D%20x)
Step-by-step explanation:
The equation is y =
. To draw the line, take two points (0, 0) and (3, 4). You can draw a straight line connecting those points.
Answer:
10 kph
Step-by-step explanation:
Find the speed of the current of the river.
20/(20 - c) + 180/(20 + c) = 8
20(20 + c) + 180(20 - c) = 8(400 - c^2)
8c^2 - 160c + 800 = 0
8(c^2 - 20c + 100) = 0
8(c - 10)^2 = 0
c = 10
The speed of the current of the river is 10 km/hour.
Answer:
.46 x 14 = 6.44
46 x 1.4 = 64.4
46 x 14 = 644
4.6 x 14 = 64.4
46 x .14 = 6.44
Step-by-step explanation:
Looks like the given limit is
![\displaystyle \lim_{n\to\infty} \left(\frac n{3n-1}\right)^{n-1}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%28%5Cfrac%20n%7B3n-1%7D%5Cright%29%5E%7Bn-1%7D)
With some simple algebra, we can rewrite
![\dfrac n{3n-1} = \dfrac13 \cdot \dfrac n{n-9} = \dfrac13 \cdot \dfrac{(n-9)+9}{n-9} = \dfrac13 \cdot \left(1 + \dfrac9{n-9}\right)](https://tex.z-dn.net/?f=%5Cdfrac%20n%7B3n-1%7D%20%3D%20%5Cdfrac13%20%5Ccdot%20%5Cdfrac%20n%7Bn-9%7D%20%3D%20%5Cdfrac13%20%5Ccdot%20%5Cdfrac%7B%28n-9%29%2B9%7D%7Bn-9%7D%20%3D%20%5Cdfrac13%20%5Ccdot%20%5Cleft%281%20%2B%20%5Cdfrac9%7Bn-9%7D%5Cright%29)
then distribute the limit over the product,
![\displaystyle \lim_{n\to\infty} \left(\frac n{3n-1}\right)^{n-1} = \lim_{n\to\infty}\left(\dfrac13\right)^{n-1} \cdot \lim_{n\to\infty}\left(1+\dfrac9{n-9}\right)^{n-1}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%28%5Cfrac%20n%7B3n-1%7D%5Cright%29%5E%7Bn-1%7D%20%3D%20%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%28%5Cdfrac13%5Cright%29%5E%7Bn-1%7D%20%5Ccdot%20%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%281%2B%5Cdfrac9%7Bn-9%7D%5Cright%29%5E%7Bn-1%7D)
The first limit is 0, since 1/3ⁿ is a positive, decreasing sequence. But before claiming the overall limit is also 0, we need to show that the second limit is also finite.
For the second limit, recall the definition of the constant, <em>e</em> :
![\displaystyle e = \lim_{n\to\infty} \left(1+\frac1n\right)^n](https://tex.z-dn.net/?f=%5Cdisplaystyle%20e%20%3D%20%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%281%2B%5Cfrac1n%5Cright%29%5En)
To make our limit resemble this one more closely, make a substitution; replace 9/(<em>n</em> - 9) with 1/<em>m</em>, so that
![\dfrac{9}{n-9} = \dfrac1m \implies 9m = n-9 \implies 9m+8 = n-1](https://tex.z-dn.net/?f=%5Cdfrac%7B9%7D%7Bn-9%7D%20%3D%20%5Cdfrac1m%20%5Cimplies%209m%20%3D%20n-9%20%5Cimplies%209m%2B8%20%3D%20n-1)
From the relation 9<em>m</em> = <em>n</em> - 9, we see that <em>m</em> also approaches infinity as <em>n</em> approaches infinity. So, the second limit is rewritten as
![\displaystyle\lim_{n\to\infty}\left(1+\dfrac9{n-9}\right)^{n-1} = \lim_{m\to\infty}\left(1+\dfrac1m\right)^{9m+8}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%281%2B%5Cdfrac9%7Bn-9%7D%5Cright%29%5E%7Bn-1%7D%20%3D%20%5Clim_%7Bm%5Cto%5Cinfty%7D%5Cleft%281%2B%5Cdfrac1m%5Cright%29%5E%7B9m%2B8%7D)
Now we apply some more properties of multiplication and limits:
![\displaystyle \lim_{m\to\infty}\left(1+\dfrac1m\right)^{9m+8} = \lim_{m\to\infty}\left(1+\dfrac1m\right)^{9m} \cdot \lim_{m\to\infty}\left(1+\dfrac1m\right)^8 \\\\ = \lim_{m\to\infty}\left(\left(1+\dfrac1m\right)^m\right)^9 \cdot \left(\lim_{m\to\infty}\left(1+\dfrac1m\right)\right)^8 \\\\ = \left(\lim_{m\to\infty}\left(1+\dfrac1m\right)^m\right)^9 \cdot \left(\lim_{m\to\infty}\left(1+\dfrac1m\right)\right)^8 \\\\ = e^9 \cdot 1^8 = e^9](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bm%5Cto%5Cinfty%7D%5Cleft%281%2B%5Cdfrac1m%5Cright%29%5E%7B9m%2B8%7D%20%3D%20%5Clim_%7Bm%5Cto%5Cinfty%7D%5Cleft%281%2B%5Cdfrac1m%5Cright%29%5E%7B9m%7D%20%5Ccdot%20%5Clim_%7Bm%5Cto%5Cinfty%7D%5Cleft%281%2B%5Cdfrac1m%5Cright%29%5E8%20%5C%5C%5C%5C%20%3D%20%5Clim_%7Bm%5Cto%5Cinfty%7D%5Cleft%28%5Cleft%281%2B%5Cdfrac1m%5Cright%29%5Em%5Cright%29%5E9%20%5Ccdot%20%5Cleft%28%5Clim_%7Bm%5Cto%5Cinfty%7D%5Cleft%281%2B%5Cdfrac1m%5Cright%29%5Cright%29%5E8%20%5C%5C%5C%5C%20%3D%20%5Cleft%28%5Clim_%7Bm%5Cto%5Cinfty%7D%5Cleft%281%2B%5Cdfrac1m%5Cright%29%5Em%5Cright%29%5E9%20%5Ccdot%20%5Cleft%28%5Clim_%7Bm%5Cto%5Cinfty%7D%5Cleft%281%2B%5Cdfrac1m%5Cright%29%5Cright%29%5E8%20%5C%5C%5C%5C%20%3D%20e%5E9%20%5Ccdot%201%5E8%20%3D%20e%5E9)
So, the overall limit is indeed 0:
![\displaystyle \lim_{n\to\infty} \left(\frac n{3n-1}\right)^{n-1} = \underbrace{\lim_{n\to\infty}\left(\dfrac13\right)^{n-1}}_0 \cdot \underbrace{\lim_{n\to\infty}\left(1+\dfrac9{n-9}\right)^{n-1}}_{e^9} = \boxed{0}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%28%5Cfrac%20n%7B3n-1%7D%5Cright%29%5E%7Bn-1%7D%20%3D%20%5Cunderbrace%7B%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%28%5Cdfrac13%5Cright%29%5E%7Bn-1%7D%7D_0%20%5Ccdot%20%5Cunderbrace%7B%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%281%2B%5Cdfrac9%7Bn-9%7D%5Cright%29%5E%7Bn-1%7D%7D_%7Be%5E9%7D%20%3D%20%5Cboxed%7B0%7D)
The answer is B. 2y+3
this is proven by multiplying the divisor of the original question and the solution