The slope form: y = mx + b
We have:
3x - 6y - 2 = 0 |add 6y to both sides
3x - 2 = 6y |divide both sides by 6
3x/6 - 2/6 = 6y/6
x/2 - 1/3 = y
y = 1/2 x - 1/3
Answer:
The idea is to transform the expression by multiplying 
with its conjugate, 
.
Step-by-step explanation:
For any real number 
and 
, 
.
The factor is irrational. However, when multiplied with its square root conjugate , the product would become rational:
.
The idea is to multiply 
by 
so as to make it easier to take the limit.
Since 
, multiplying the expression by this fraction would not change the value of the original expression.
![\begin{aligned} & \lim\limits_{x \to \infty} \sqrt{x} \, (\sqrt{x + 1} - \sqrt{x}) \\ &= \lim\limits_{x \to \infty} \left[\sqrt{x} \, (\sqrt{x + 1} - \sqrt{x})\cdot \frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}\right] \\ &= \lim\limits_{x \to \infty} \frac{\sqrt{x}\, ((x + 1) - x)}{\sqrt{x + 1} + \sqrt{x}} \\ &= \lim\limits_{x \to \infty} \frac{\sqrt{x}}{\sqrt{x + 1}+ \sqrt{x}}\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%20%26%20%5Clim%5Climits_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Csqrt%7Bx%7D%20%5C%2C%20%28%5Csqrt%7Bx%20%2B%201%7D%20-%20%5Csqrt%7Bx%7D%29%20%5C%5C%20%26%3D%20%5Clim%5Climits_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cleft%5B%5Csqrt%7Bx%7D%20%5C%2C%20%28%5Csqrt%7Bx%20%2B%201%7D%20-%20%5Csqrt%7Bx%7D%29%5Ccdot%20%5Cfrac%7B%5Csqrt%7Bx%20%2B%201%7D%20%2B%20%5Csqrt%7Bx%7D%7D%7B%5Csqrt%7Bx%20%2B%201%7D%20%2B%20%5Csqrt%7Bx%7D%7D%5Cright%5D%20%5C%5C%20%26%3D%20%5Clim%5Climits_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B%5Csqrt%7Bx%7D%5C%2C%20%28%28x%20%2B%201%29%20-%20x%29%7D%7B%5Csqrt%7Bx%20%2B%201%7D%20%2B%20%5Csqrt%7Bx%7D%7D%20%5C%5C%20%26%3D%20%5Clim%5Climits_%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B%5Csqrt%7Bx%7D%7D%7B%5Csqrt%7Bx%20%2B%201%7D%2B%20%5Csqrt%7Bx%7D%7D%5Cend%7Baligned%7D)
.
The order of 
in both the numerator and the denominator are now both 
. Hence, dividing both the numerator and the denominator by 
(same as 
) would ensure that all but the constant terms would approach 
under this limit:
.
By continuity:
.
Answer:
the "a" is the initial value so the starting amount/ y- intercept
the "b" is the growth/decay factor
