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aivan3 [116]
4 years ago
11

Evaluate the limit, if it exists. (if an answer does not exist, enter dne.)lim h → 0 (x + h)3 − x3h

Mathematics
1 answer:
Natalka [10]4 years ago
7 0
Presumably, the limit is

\displaystyle\lim_{h\to0}\frac{(x+h)^3-x^3}h

Now, if you're familiar with the definition of the derivatives, you'll notice that this is the limit form of the derivative of the function f(x)=x^3, which you may also know to be 3x^2. But let's assume you don't know that just yet, and that it's actually the result you intend to find.

Expand the numerator:

\dfrac{(x+h)^3-x^3}h=\dfrac{(x^3+3x^2h+3xh^2+h^3)-x^3}h=\dfrac{3x^2h+3xh+h^3}h

Now, when h\neq0, we can divide through by the lowest power of h. We can do this because we're considering the limit as h is *approaching* 0, and not when it actually takes on the value of h=0.

\dfrac{(x+h)^3-x^3}h=3x^2+3xh+h^2

Now, as h\to0, we can see only the leading term remains, so that

\displaystyle\lim_{h\to0}\frac{(x+h)^3-x^3}h=3x^2

as expected.
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Express 2x^2+8x+3 in the form 2(x+p)^2+q
e-lub [12.9K]

Answer:

2(x + 2)² - 5

Step-by-step explanation:

Given

2x² + 8x + 3

To obtain the required form use the method of completing the square.

The coefficient of the x² term must be 1, thus factor 2 out of 2x² + 8x

= 2(x² + 4x) + 3

add/subtract ( half the coefficient of the x- term)² to x² + 4x

= 2(x² + 2(2)x + 4 - 4) + 3

= 2(x + 2)² - 8 + 3

= 2(x + 2)² - 5

with p = 2 and q = - 5

4 0
4 years ago
Any number that CAN be divided by 2 without having remainder is considered an _______ number
azamat

Step-by-step explanation:

Any number that can be divided by 2 without having remainder is considered an even number.

I hope it helped U

stay safe stay happy

8 0
3 years ago
Find the perimeter of a rectangle with a length of 6x+3 and a width of -2x-5.
blsea [12.9K]
The simple way of calculating a perimeter is to add all of the sides.
For a rectangle this would equal 2 x length + 2 x width
P = 2(6x + 3) + 2(-2x - 5)
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= 8x - 4

At a higher level both the length and width must be greater than zero (= zero is a trivial rectangle)
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-2x - 5 > 0
2x + 5 < 0 (multiplying by -1 reverses the inequality)
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This rectangle cannot exist as x cannot be < -2.5 and > -0.5 at the same time!
8 0
3 years ago
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Temka [501]
Let x=number 1 let y=number2
there sum is equal to 70
then

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substituting

x+y=70

y=70-x
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p=xy
substituting
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Aleks04 [339]

Answer:

Step-by-step explanation:

The Area of rectangle =is express as [ 5 x² + 10 x ] sq unit Answer. 5.0

7 0
4 years ago
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