Answer: y = (xm)/3 + b
Step-by-step explanation:
1. Multiply m on both sides
x × m = 3(y - b)
2. Divide by 3 on both sides
(xm)/3 = y - b
3. Add b on both sides
y = (xm)/3 + b
Answer:
In general, when y = f(x), you are substituting a permissible x value into function f & its calculated 'output' is the value for y. Then (x,y) is an ordered pair, i.e. coordinates of a point on the graph. ... So our ordered pair is (-14,11).
For this case, the first thing we are going to do is rewrite the expression.
We have then:
(18) / root (8)
(18) / root (2 * 4)
(18) / 2 * root (2)
(9) / root (2)
Thus, the values of a and b are:
a = 9
b = 2
Answer:
a = 9
b = 2
Answer:
Step-by-step explanation:
Considering the expression
![\lim _{x\to \infty \:}\frac{\sqrt{9x^2+x+1}-\sqrt{4x^2+2x+1}}{x+1}](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cfrac%7B%5Csqrt%7B9x%5E2%2Bx%2B1%7D-%5Csqrt%7B4x%5E2%2B2x%2B1%7D%7D%7Bx%2B1%7D)
Steps to solve
![\lim _{x\to \infty \:}\frac{\sqrt{9x^2+x+1}-\sqrt{4x^2+2x+1}}{x+1}](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cfrac%7B%5Csqrt%7B9x%5E2%2Bx%2B1%7D-%5Csqrt%7B4x%5E2%2B2x%2B1%7D%7D%7Bx%2B1%7D)
![\mathrm{Divide\:by\:highest\:denominator\:power:}\:\frac{\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}-\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}}{1+\frac{1}{x}}](https://tex.z-dn.net/?f=%5Cmathrm%7BDivide%5C%3Aby%5C%3Ahighest%5C%3Adenominator%5C%3Apower%3A%7D%5C%3A%5Cfrac%7B%5Csqrt%7B9%2B%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D-%5Csqrt%7B4%2B%5Cfrac%7B2%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%7D%7B1%2B%5Cfrac%7B1%7D%7Bx%7D%7D)
![\lim _{x\to \infty \:}\left(\frac{\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}-\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}}{1+\frac{1}{x}}\right)](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B%5Csqrt%7B9%2B%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D-%5Csqrt%7B4%2B%5Cfrac%7B2%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%7D%7B1%2B%5Cfrac%7B1%7D%7Bx%7D%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)
![\mathrm{With\:the\:exception\:of\:indeterminate\:form}](https://tex.z-dn.net/?f=%5Cmathrm%7BWith%5C%3Athe%5C%3Aexception%5C%3Aof%5C%3Aindeterminate%5C%3Aform%7D)
![\frac{\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}-\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}\right)}{\lim _{x\to \infty \:}\left(1+\frac{1}{x}\right)}.....[1]](https://tex.z-dn.net/?f=%5Cfrac%7B%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Csqrt%7B9%2B%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D-%5Csqrt%7B4%2B%5Cfrac%7B2%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%5Cright%29%7D%7B%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%281%2B%5Cfrac%7B1%7D%7Bx%7D%5Cright%29%7D.....%5B1%5D)
As
![\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}-\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}\right)=1](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Csqrt%7B9%2B%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D-%5Csqrt%7B4%2B%5Cfrac%7B2%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%5Cright%29%3D1)
Solving
![\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}-\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}\right)....[A]](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Csqrt%7B9%2B%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D-%5Csqrt%7B4%2B%5Cfrac%7B2%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%5Cright%29....%5BA%5D)
![\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)
![\mathrm{With\:the\:exception\:of\:indeterminate\:form}](https://tex.z-dn.net/?f=%5Cmathrm%7BWith%5C%3Athe%5C%3Aexception%5C%3Aof%5C%3Aindeterminate%5C%3Aform%7D)
![\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}\right)-\lim _{x\to \infty \:}\left(\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}\right)](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Csqrt%7B9%2B%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%5Cright%29-%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Csqrt%7B4%2B%5Cfrac%7B2%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%5Cright%29)
Also
![\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}\right)=3](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Csqrt%7B9%2B%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%5Cright%29%3D3)
Solving
![\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}\right)......[B]](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Csqrt%7B9%2B%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%5Cright%29......%5BB%5D)
![\lim _{x\to a}\left[f\left(x\right)\right]^b=\left[\lim _{x\to a}f\left(x\right)\right]^b](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20a%7D%5Cleft%5Bf%5Cleft%28x%5Cright%29%5Cright%5D%5Eb%3D%5Cleft%5B%5Clim%20_%7Bx%5Cto%20a%7Df%5Cleft%28x%5Cright%29%5Cright%5D%5Eb)
![\mathrm{With\:the\:exception\:of\:indeterminate\:form}](https://tex.z-dn.net/?f=%5Cmathrm%7BWith%5C%3Athe%5C%3Aexception%5C%3Aof%5C%3Aindeterminate%5C%3Aform%7D)
![\sqrt{\lim _{x\to \infty \:}\left(9+\lim _{x\to \infty \:}\left(\frac{1}{x}+\lim _{x\to \infty \:}\left(\frac{1}{x^2}\right)\right)\right)}](https://tex.z-dn.net/?f=%5Csqrt%7B%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%289%2B%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B1%7D%7Bx%7D%2B%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B1%7D%7Bx%5E2%7D%5Cright%29%5Cright%29%5Cright%29%7D)
![\lim _{x\to \infty \:}\left(9\right)=9](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%289%5Cright%29%3D9)
![\lim _{x\to \infty \:}\left(\frac{1}{x}\right)=0](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B1%7D%7Bx%7D%5Cright%29%3D0)
![\lim _{x\to \infty \:}\left(\frac{1}{x^2}\right)=0](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Cfrac%7B1%7D%7Bx%5E2%7D%5Cright%29%3D0)
So, Equation [B] becomes
⇒ ![\sqrt{9+0+0}](https://tex.z-dn.net/?f=%5Csqrt%7B9%2B0%2B0%7D)
⇒ ![3](https://tex.z-dn.net/?f=3)
Similarly, we can find
![\lim _{x\to \infty \:}\left(\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}\right)=2](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Csqrt%7B4%2B%5Cfrac%7B2%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%5Cright%29%3D2)
So, Equation [A] becomes
⇒ ![3-2](https://tex.z-dn.net/?f=3-2)
⇒ 1
Also
![\lim _{x\to \infty \:}\left(1+\frac{1}{x}\right)=1](https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%281%2B%5Cfrac%7B1%7D%7Bx%7D%5Cright%29%3D1)
Thus, equation becomes
![\frac{\lim _{x\to \infty \:}\left(\sqrt{9+\frac{1}{x}+\frac{1}{x^2}}-\sqrt{4+\frac{2}{x}+\frac{1}{x^2}}\right)}{\lim _{x\to \infty \:}\left(1+\frac{1}{x}\right)}=\frac{1}{1}=1](https://tex.z-dn.net/?f=%5Cfrac%7B%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%28%5Csqrt%7B9%2B%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D-%5Csqrt%7B4%2B%5Cfrac%7B2%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%5Cright%29%7D%7B%5Clim%20_%7Bx%5Cto%20%5Cinfty%20%5C%3A%7D%5Cleft%281%2B%5Cfrac%7B1%7D%7Bx%7D%5Cright%29%7D%3D%5Cfrac%7B1%7D%7B1%7D%3D1)
Therefore,
Keywords: limit
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