Answer:
1/2 gallons
Step-by-step explanation:
since 1/3of the tank is already full and another 4 gallons fills it up
4 gallons= 2/3 of the tank or the tank is 6 gallons total
75% is equal to 3/4
6*3/4=4 1/2 gallons
since it is 1/3 or 4 gallons full
4 1/2 -4= 1/2
Answer: Third option
Step-by-step explanation:
Remember that:
![\sqrt[n]{x^n}=x](https://tex.z-dn.net/?f=%5Csqrt%5Bn%5D%7Bx%5En%7D%3Dx)
And the Product of powers property:
The expression is:
![\sqrt[4]{\frac{3}{2x}}](https://tex.z-dn.net/?f=%5Csqrt%5B4%5D%7B%5Cfrac%7B3%7D%7B2x%7D%7D)
To simplify this expression, you need to multiply the denominator and the numerator by 
. Then:
![\frac{\sqrt[4]{3}}{\sqrt[4]{2x}}=\frac{\sqrt[4]{3(2^3x^3)}}{\sqrt[4]{2x(2^3x^3)}}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%5B4%5D%7B3%7D%7D%7B%5Csqrt%5B4%5D%7B2x%7D%7D%3D%5Cfrac%7B%5Csqrt%5B4%5D%7B3%282%5E3x%5E3%29%7D%7D%7B%5Csqrt%5B4%5D%7B2x%282%5E3x%5E3%29%7D%7D)
Simplifiying, you get:
![\frac{\sqrt[4]{3(8x^3)}}{\sqrt[4]{2x(2^3x^3)}}=\frac{\sqrt[4]{24x^3}}{\sqrt[4]{2^4x^4}}=\frac{\sqrt[4]{24x^3}}{2x}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%5B4%5D%7B3%288x%5E3%29%7D%7D%7B%5Csqrt%5B4%5D%7B2x%282%5E3x%5E3%29%7D%7D%3D%5Cfrac%7B%5Csqrt%5B4%5D%7B24x%5E3%7D%7D%7B%5Csqrt%5B4%5D%7B2%5E4x%5E4%7D%7D%3D%5Cfrac%7B%5Csqrt%5B4%5D%7B24x%5E3%7D%7D%7B2x%7D)
<span>1. We analyze the limit by approaching it from both the left and the right.
From the left: f(x) = x + 10 (for x < 8), as x --> 8, f(x) --> 18
From the right: f(x) = 10 - x (for x >= 8), as x --> 8, f(x) --> 2
Since the limits on either side do not converge to the same point, the limit does not exist (this is choice C).
2. </span>Using a similar approaching as in #1:
<span><span>From the left: f(x) = 5 - x (for x < 5), as x --> 5, f(x) --> 0
At x = 5 itself: f(x) = 8
From the right: f(x) = x + 3 (for x > 5), as x --> 5, f(x) --> 8</span>
Although the value at x = 5 matches with the limit when approaching from the right, the limit when approaching from the left doesn't match, so the limit does not exist (choice D).
3. </span><span><span>From the left: f(x) = 5x - 9 (for x < 0), as x --> 0, f(x) --> -9
From the right: f(x) = |2 - x| (for x >= 0), as x --> 0, f(x) --> 2
</span>Again, since the limits when approaching from the left and right don't match, the limit does not exist. (This is Choice D).
4. lim 1/(x - 4) as x -->4-
If we are approaching x = 4 from the left, we can test values such as 3, 3.9, 3.99, 3.999, approaching 4. For x = 3, f(x) = -1. For x = 3.9, f(x) = -10. For x = 3.99, f(x) = -100. For x = 3.999, f(x) = -1000. This shows that the value continues to go towards negative infinity.
If we were to graph these 4 points on the Cartesian plane, it would also show a curve to slopes downwards to negative infinity, with the vertical asymptote at x = 4. The correct answer is Choice C) -∞ ; x = 4.
5. </span>f(x) = (x+1)(x-1) / [(x+1)(x-2)] is an example of a function with both a removable and non-removable discontinuity.
In this case, because x+1 cancels out from the numerator and denominator, it results in a hollow or missing point (removable) discontinuity at x = -1. This means that the limit still exists as x --> -1. On the other hand, x = 2 is a non-removable discontinuity, since it cannot be cancelled out, and it will be an asymptote.
