2/5 ÷ x = 4/5
multiply both sides by x to get it out of the denominator.
2/5 = 4/5x
now divide each side by 4/5 to get x by itself.
2/5 ÷ 4/5 = x
divide fractions by multiplying the reciprocal of the divisor.
2/5x × 5/4 = x
10/20 = x
1/2 = x
So 2/5 divided by 1/2 =4/5
Answer:
The answer is y = x + 38
PLEASE MARK BRAINLIEST!
AND THIS WAS ONLY 5 POINTS
<h2>
4. найдите пределы:</h2>
![1)\:\:\:\:\:\:\:\:\lim _{n\to \infty \:}\left(\frac{2n-3}{5+4n}\right)](https://tex.z-dn.net/?f=1%29%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B2n-3%7D%7B5%2B4n%7D%5Cright%29)
Пошаговое решение:
![\lim _{n\to \infty \:}\left(\frac{2n-3}{5+4n}\right)](https://tex.z-dn.net/?f=%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B2n-3%7D%7B5%2B4n%7D%5Cright%29)
Разделите на высшую знаменательную силу 'n'
![=\frac{\frac{2n}{n}-\frac{3}{n}}{\frac{5}{n}+\frac{4n}{n}}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B%5Cfrac%7B2n%7D%7Bn%7D-%5Cfrac%7B3%7D%7Bn%7D%7D%7B%5Cfrac%7B5%7D%7Bn%7D%2B%5Cfrac%7B4n%7D%7Bn%7D%7D)
![=\frac{2-\frac{3}{n}}{\frac{5}{n}+4}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2-%5Cfrac%7B3%7D%7Bn%7D%7D%7B%5Cfrac%7B5%7D%7Bn%7D%2B4%7D)
так
![=\lim _{n\to \infty \:}\left(\frac{2-\frac{3}{n}}{\frac{5}{n}+4}\right)](https://tex.z-dn.net/?f=%3D%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B2-%5Cfrac%7B3%7D%7Bn%7D%7D%7B%5Cfrac%7B5%7D%7Bn%7D%2B4%7D%5Cright%29)
![\lim _{x\to a}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{\lim _{x\to a}f\left(x\right)}{\lim _{x\to a}g\left(x\right)},\:\quad \lim _{x\to a}g\left(x\right)\ne 0](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20a%7D%5Cleft%5B%5Cfrac%7Bf%5Cleft%28x%5Cright%29%7D%7Bg%5Cleft%28x%5Cright%29%7D%5Cright%5D%3D%5Cfrac%7B%5Clim%20_%7Bx%5Cto%20a%7Df%5Cleft%28x%5Cright%29%7D%7B%5Clim%20_%7Bx%5Cto%20a%7Dg%5Cleft%28x%5Cright%29%7D%2C%5C%3A%5Cquad%20%5Clim%20_%7Bx%5Cto%20a%7Dg%5Cleft%28x%5Cright%29%5Cne%200)
![=\frac{\lim _{n\to \infty \:}\left(2-\frac{3}{n}\right)}{\lim _{n\to \infty \:}\left(\frac{5}{n}+4\right)}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%282-%5Cfrac%7B3%7D%7Bn%7D%5Cright%29%7D%7B%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B5%7D%7Bn%7D%2B4%5Cright%29%7D)
так как
![\lim _{n\to \infty \:}\left(2-\frac{3}{n}\right)](https://tex.z-dn.net/?f=%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%282-%5Cfrac%7B3%7D%7Bn%7D%5Cright%29)
![\lim _{x\to a}\left[f\left(x\right)\pm g\left(x\right)\right]=\lim _{x\to a}f\left(x\right)\pm \lim _{x\to a}g\left(x\right)](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20a%7D%5Cleft%5Bf%5Cleft%28x%5Cright%29%5Cpm%20g%5Cleft%28x%5Cright%29%5Cright%5D%3D%5Clim%20_%7Bx%5Cto%20a%7Df%5Cleft%28x%5Cright%29%5Cpm%20%5Clim%20_%7Bx%5Cto%20a%7Dg%5Cleft%28x%5Cright%29)
![=\lim _{n\to \infty \:}\left(\left(2\right)-\lim _{n\to \infty \:}\left(\frac{3}{n}\right)\right)](https://tex.z-dn.net/?f=%3D%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cleft%282%5Cright%29-%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B3%7D%7Bn%7D%5Cright%29%5Cright%29)
![\lim _{n\to \infty \:}\left(2\right)=2](https://tex.z-dn.net/?f=%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%282%5Cright%29%3D2)
![\lim _{n\to \infty \:}\left(2\right)=2](https://tex.z-dn.net/?f=%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%282%5Cright%29%3D2)
![=2-0](https://tex.z-dn.net/?f=%3D2-0)
![=2](https://tex.z-dn.net/?f=%3D2)
также
![\lim _{n\to \infty \:}\left(\frac{5}{n}+4\right)=4](https://tex.z-dn.net/?f=%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B5%7D%7Bn%7D%2B4%5Cright%29%3D4)
Следовательно
![=\lim _{n\to \infty \:}\left(\frac{2-\frac{3}{n}}{\frac{5}{n}+4}\right)](https://tex.z-dn.net/?f=%3D%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B2-%5Cfrac%7B3%7D%7Bn%7D%7D%7B%5Cfrac%7B5%7D%7Bn%7D%2B4%7D%5Cright%29)
![=\frac{2}{4}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2%7D%7B4%7D)
![=\frac{1}{2}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B2%7D)
∵ ![\lim _{n\to \infty \:}\left(\frac{2n-3}{5+4n}\right)=\frac{1}{2}](https://tex.z-dn.net/?f=%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B2n-3%7D%7B5%2B4n%7D%5Cright%29%3D%5Cfrac%7B1%7D%7B2%7D)
![2)\:\:\:\:\:\:\:\:\:\:\lim _{n\to \infty \:}\left(\frac{4n^2-3n+5}{7-2n^2}\right)](https://tex.z-dn.net/?f=2%29%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B4n%5E2-3n%2B5%7D%7B7-2n%5E2%7D%5Cright%29)
Пошаговое решение:
![\lim _{n\to \infty \:}\left(\frac{4n^2-3n+5}{7-2n^2}\right)](https://tex.z-dn.net/?f=%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B4n%5E2-3n%2B5%7D%7B7-2n%5E2%7D%5Cright%29)
Разделите на высшую знаменательную силу 'n²'
![=\lim _{n\to \infty \:}\left(\frac{4-\frac{3}{n}+\frac{5}{n^2}}{\frac{7}{n^2}-2}\right)](https://tex.z-dn.net/?f=%3D%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B4-%5Cfrac%7B3%7D%7Bn%7D%2B%5Cfrac%7B5%7D%7Bn%5E2%7D%7D%7B%5Cfrac%7B7%7D%7Bn%5E2%7D-2%7D%5Cright%29)
![\lim _{x\to a}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{\lim _{x\to a}f\left(x\right)}{\lim _{x\to a}g\left(x\right)},\:\quad \lim _{x\to a}g\left(x\right)\ne 0](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20a%7D%5Cleft%5B%5Cfrac%7Bf%5Cleft%28x%5Cright%29%7D%7Bg%5Cleft%28x%5Cright%29%7D%5Cright%5D%3D%5Cfrac%7B%5Clim%20_%7Bx%5Cto%20a%7Df%5Cleft%28x%5Cright%29%7D%7B%5Clim%20_%7Bx%5Cto%20a%7Dg%5Cleft%28x%5Cright%29%7D%2C%5C%3A%5Cquad%20%5Clim%20_%7Bx%5Cto%20a%7Dg%5Cleft%28x%5Cright%29%5Cne%200)
![=\frac{\lim _{n\to \infty \:}\left(4-\frac{3}{n}+\frac{5}{n^2}\right)}{\lim _{n\to \infty \:}\left(\frac{7}{n^2}-2\right)}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%284-%5Cfrac%7B3%7D%7Bn%7D%2B%5Cfrac%7B5%7D%7Bn%5E2%7D%5Cright%29%7D%7B%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B7%7D%7Bn%5E2%7D-2%5Cright%29%7D)
так как
![\lim _{n\to \infty \:}\left(4-\frac{3}{n}+\frac{5}{n^2}\right)=4](https://tex.z-dn.net/?f=%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%284-%5Cfrac%7B3%7D%7Bn%7D%2B%5Cfrac%7B5%7D%7Bn%5E2%7D%5Cright%29%3D4)
и
![\lim _{n\to \infty \:}\left(\frac{7}{n^2}-2\right)=-2](https://tex.z-dn.net/?f=%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B7%7D%7Bn%5E2%7D-2%5Cright%29%3D-2)
так
![=\lim _{n\to \infty \:}\left(\frac{4-\frac{3}{n}+\frac{5}{n^2}}{\frac{7}{n^2}-2}\right)](https://tex.z-dn.net/?f=%3D%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B4-%5Cfrac%7B3%7D%7Bn%7D%2B%5Cfrac%7B5%7D%7Bn%5E2%7D%7D%7B%5Cfrac%7B7%7D%7Bn%5E2%7D-2%7D%5Cright%29)
![=\frac{4}{-2}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B4%7D%7B-2%7D)
![=-2](https://tex.z-dn.net/?f=%3D-2)
∵ ![\lim _{n\to \infty \:}\left(\frac{4n^2-3n+5}{7-2n^2}\right)=-2](https://tex.z-dn.net/?f=%5Clim%20_%7Bn%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B4n%5E2-3n%2B5%7D%7B7-2n%5E2%7D%5Cright%29%3D-2)
<h2>
5.</h2>
1)
![\lim \:_{x\to \:1\:\:}\left(\frac{4x-2}{2x^2+3x+7}\right)](https://tex.z-dn.net/?f=%5Clim%20%5C%3A_%7Bx%5Cto%20%5C%3A1%5C%3A%5C%3A%7D%5Cleft%28%5Cfrac%7B4x-2%7D%7B2x%5E2%2B3x%2B7%7D%5Cright%29)
Пошаговое решение:
![\lim _{x\to \:1\:\:}\left(\frac{4x-2}{2x^2+3x+7}\right)](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A1%5C%3A%5C%3A%7D%5Cleft%28%5Cfrac%7B4x-2%7D%7B2x%5E2%2B3x%2B7%7D%5Cright%29)
Положил x = 1
![=\frac{4\cdot \:1-2}{2\cdot \:1^2+3\cdot \:1+7}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B4%5Ccdot%20%5C%3A1-2%7D%7B2%5Ccdot%20%5C%3A1%5E2%2B3%5Ccdot%20%5C%3A1%2B7%7D)
![=\frac{2}{2\cdot \:1+3\cdot \:1+7}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2%7D%7B2%5Ccdot%20%5C%3A1%2B3%5Ccdot%20%5C%3A1%2B7%7D)
∵ ![2\cdot \:1+3\cdot \:1+7=12](https://tex.z-dn.net/?f=2%5Ccdot%20%5C%3A1%2B3%5Ccdot%20%5C%3A1%2B7%3D12)
![=\frac{1}{6}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B6%7D)
∵ ![\lim _{x\to \:1}\left(\frac{4x-2}{2x^2+3x+7}\right)=\frac{1}{6}](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A1%7D%5Cleft%28%5Cfrac%7B4x-2%7D%7B2x%5E2%2B3x%2B7%7D%5Cright%29%3D%5Cfrac%7B1%7D%7B6%7D)
2)
![\lim _{x\to \:1\:\:}\left(\frac{2x^2-5x+3}{x^2-1}\right)](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A1%5C%3A%5C%3A%7D%5Cleft%28%5Cfrac%7B2x%5E2-5x%2B3%7D%7Bx%5E2-1%7D%5Cright%29)
Пошаговое решение:
![\lim _{x\to \:1}\left(\frac{2x^2-5x+3}{x^2-1}\right)](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A1%7D%5Cleft%28%5Cfrac%7B2x%5E2-5x%2B3%7D%7Bx%5E2-1%7D%5Cright%29)
Решить
![\frac{2x^2-5x+3}{x^2-1}](https://tex.z-dn.net/?f=%5Cfrac%7B2x%5E2-5x%2B3%7D%7Bx%5E2-1%7D)
так как
![2x^2-5x+3:\quad \left(x-1\right)\left(2x-3\right)](https://tex.z-dn.net/?f=2x%5E2-5x%2B3%3A%5Cquad%20%5Cleft%28x-1%5Cright%29%5Cleft%282x-3%5Cright%29)
![x^2-1:\quad \left(x+1\right)\left(x-1\right)](https://tex.z-dn.net/?f=x%5E2-1%3A%5Cquad%20%5Cleft%28x%2B1%5Cright%29%5Cleft%28x-1%5Cright%29)
так
![=\frac{\left(x-1\right)\left(2x-3\right)}{\left(x+1\right)\left(x-1\right)}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B%5Cleft%28x-1%5Cright%29%5Cleft%282x-3%5Cright%29%7D%7B%5Cleft%28x%2B1%5Cright%29%5Cleft%28x-1%5Cright%29%7D)
![=\frac{2x-3}{x+1}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2x-3%7D%7Bx%2B1%7D)
Итак, уравнение:
![=\lim _{x\to \:1}\left(\frac{2x-3}{x+1}\right)](https://tex.z-dn.net/?f=%3D%5Clim%20_%7Bx%5Cto%20%5C%3A1%7D%5Cleft%28%5Cfrac%7B2x-3%7D%7Bx%2B1%7D%5Cright%29)
Положил x = 1
![=\frac{2\cdot \:1-3}{1+1}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2%5Ccdot%20%5C%3A1-3%7D%7B1%2B1%7D)
![=-\frac{1}{2}](https://tex.z-dn.net/?f=%3D-%5Cfrac%7B1%7D%7B2%7D)
∵ ![\lim _{x\to \:1}\left(\frac{2x^2-5x+3}{x^2-1}\right)=-\frac{1}{2}](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A1%7D%5Cleft%28%5Cfrac%7B2x%5E2-5x%2B3%7D%7Bx%5E2-1%7D%5Cright%29%3D-%5Cfrac%7B1%7D%7B2%7D)
3)
![\lim _{x\to \:3}\left(\frac{\sqrt{x+5}-3}{x-3}\right)](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A3%7D%5Cleft%28%5Cfrac%7B%5Csqrt%7Bx%2B5%7D-3%7D%7Bx-3%7D%5Cright%29)
Пошаговое решение:
![\lim _{x\to \:3}\left(\frac{\sqrt{x+5}-3}{x-3}\right)](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A3%7D%5Cleft%28%5Cfrac%7B%5Csqrt%7Bx%2B5%7D-3%7D%7Bx-3%7D%5Cright%29)
![\mathrm{If\:}\lim _{x\to a-}f\left(x\right)\ne \lim _{x\to a+}f\left(x\right)\mathrm{\:then\:the\:limit\:does\:not\:exist}](https://tex.z-dn.net/?f=%5Cmathrm%7BIf%5C%3A%7D%5Clim%20_%7Bx%5Cto%20a-%7Df%5Cleft%28x%5Cright%29%5Cne%20%5Clim%20_%7Bx%5Cto%20a%2B%7Df%5Cleft%28x%5Cright%29%5Cmathrm%7B%5C%3Athen%5C%3Athe%5C%3Alimit%5C%3Adoes%5C%3Anot%5C%3Aexist%7D)
так как
![\lim _{x\to \:3+}\left(\frac{\sqrt{x+5}-3}{x-3}\right)=-\infty \:](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A3%2B%7D%5Cleft%28%5Cfrac%7B%5Csqrt%7Bx%2B5%7D-3%7D%7Bx-3%7D%5Cright%29%3D-%5Cinfty%20%5C%3A)
и
![\lim _{x\to \:3-}\left(\frac{\sqrt{x+5}-3}{x-3}\right)=\infty \:](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A3-%7D%5Cleft%28%5Cfrac%7B%5Csqrt%7Bx%2B5%7D-3%7D%7Bx-3%7D%5Cright%29%3D%5Cinfty%20%5C%3A)
так
∵ diverges mean 'расходится'
4)
![\lim _{x\to \:0\:\:}\left(\frac{sin\:4x}{x}\right)](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A0%5C%3A%5C%3A%7D%5Cleft%28%5Cfrac%7Bsin%5C%3A4x%7D%7Bx%7D%5Cright%29)
Пошаговое решение:
![\lim _{x\to \:0}\left(\frac{\sin \left(4x\right)}{x}\right)](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A0%7D%5Cleft%28%5Cfrac%7B%5Csin%20%5Cleft%284x%5Cright%29%7D%7Bx%7D%5Cright%29)
Apply L'Hopital's Rule
![=\lim _{x\to \:0}\left(\frac{\cos \left(4x\right)\cdot \:4}{1}\right)](https://tex.z-dn.net/?f=%3D%5Clim%20_%7Bx%5Cto%20%5C%3A0%7D%5Cleft%28%5Cfrac%7B%5Ccos%20%5Cleft%284x%5Cright%29%5Ccdot%20%5C%3A4%7D%7B1%7D%5Cright%29)
![=\lim _{x\to \:0}\left(4\cos \left(4x\right)\right)](https://tex.z-dn.net/?f=%3D%5Clim%20_%7Bx%5Cto%20%5C%3A0%7D%5Cleft%284%5Ccos%20%5Cleft%284x%5Cright%29%5Cright%29)
Положил x = 0
![=4\cos \left(4\cdot \:0\right)](https://tex.z-dn.net/?f=%3D4%5Ccos%20%5Cleft%284%5Ccdot%20%5C%3A0%5Cright%29)
∵ ![4\cos \left(4\cdot \:0\right):\quad 4](https://tex.z-dn.net/?f=4%5Ccos%20%5Cleft%284%5Ccdot%20%5C%3A0%5Cright%29%3A%5Cquad%204)
5)
![\lim \:_{x\to \:\infty \:\:}\left(\frac{2x^2-5x+3}{x^2-1}\right)](https://tex.z-dn.net/?f=%5Clim%20%5C%3A_%7Bx%5Cto%20%5C%3A%5Cinfty%20%5C%3A%5C%3A%7D%5Cleft%28%5Cfrac%7B2x%5E2-5x%2B3%7D%7Bx%5E2-1%7D%5Cright%29)
Пошаговое решение:
Check the attached diagram for the solution of this question.
(Проверьте прилагаемую диаграмму для решения этого вопроса.)
What are you doing if your graphing the points for the first one you start at -7 on the y axis and go up 3 and over tho the left by 1
For the second one you start at 1 on the y axis and go up 3 over the right 1