What is 675 divided by 12 long division...lol I’m TIRED.help...

n200080 [17]

Looks like the given limit is

$\displaystyle \lim_{n\to\infty} \left(\frac n{3n-1}\right)^{n-1}$

With some simple algebra, we can rewrite

$\dfrac n{3n-1} = \dfrac13 \cdot \dfrac n{n-9} = \dfrac13 \cdot \dfrac{(n-9)+9}{n-9} = \dfrac13 \cdot \left(1 + \dfrac9{n-9}\right)$

then distribute the limit over the product,

$\displaystyle \lim_{n\to\infty} \left(\frac n{3n-1}\right)^{n-1} = \lim_{n\to\infty}\left(\dfrac13\right)^{n-1} \cdot \lim_{n\to\infty}\left(1+\dfrac9{n-9}\right)^{n-1}$

The first limit is 0, since 1/3ⁿ is a positive, decreasing sequence. But before claiming the overall limit is also 0, we need to show that the second limit is also finite.

For the second limit, recall the definition of the constant, e :

$\displaystyle e = \lim_{n\to\infty} \left(1+\frac1n\right)^n$

To make our limit resemble this one more closely, make a substitution; replace 9/(n - 9) with 1/m, so that

$\dfrac{9}{n-9} = \dfrac1m \implies 9m = n-9 \implies 9m+8 = n-1$

From the relation 9m = n - 9, we see that m also approaches infinity as n approaches infinity. So, the second limit is rewritten as

$\displaystyle\lim_{n\to\infty}\left(1+\dfrac9{n-9}\right)^{n-1} = \lim_{m\to\infty}\left(1+\dfrac1m\right)^{9m+8}$

Now we apply some more properties of multiplication and limits:

$\displaystyle \lim_{m\to\infty}\left(1+\dfrac1m\right)^{9m+8} = \lim_{m\to\infty}\left(1+\dfrac1m\right)^{9m} \cdot \lim_{m\to\infty}\left(1+\dfrac1m\right)^8 \\\\ = \lim_{m\to\infty}\left(\left(1+\dfrac1m\right)^m\right)^9 \cdot \left(\lim_{m\to\infty}\left(1+\dfrac1m\right)\right)^8 \\\\ = \left(\lim_{m\to\infty}\left(1+\dfrac1m\right)^m\right)^9 \cdot \left(\lim_{m\to\infty}\left(1+\dfrac1m\right)\right)^8 \\\\ = e^9 \cdot 1^8 = e^9$

So, the overall limit is indeed 0:

$\displaystyle \lim_{n\to\infty} \left(\frac n{3n-1}\right)^{n-1} = \underbrace{\lim_{n\to\infty}\left(\dfrac13\right)^{n-1}}_0 \cdot \underbrace{\lim_{n\to\infty}\left(1+\dfrac9{n-9}\right)^{n-1}}_{e^9} = \boxed{0}$

7 0

3 years ago

Simplify the expression 1+2*[(3+1)*5+3]

Lady bird [3.3K]

1 + 2 *[(3+1)*5+3]

= 1 + 2*[(4)*5+3]

= 1 + 2*[20+3]

= 1 + 2 * 23

= 1 + 46

= 47

Answer

c. 47

7 0

2 years ago

Read 2 more answers

An object is translated by (x-2, y-6). If one point in the pre-image has the coordinates (-3, 7,) what would be the coordinates

Katen [24]

Answer:

(- 5, 1 )

Step-by-step explanation:

Given the translation rule

(x, y ) → (x - 2, y - 6 )

This means subtract 2 from the original x- coordinate and subtract 6 from the original y- coordinate, that is

(- 3, 7 ) → (- 3 - 2, 7 - 6 ) → (- 5, 1 )

5 0

3 years ago

Which situation can be represented by the inequality 4x − 25 < 125?

GaryK [48]

First add 25 to both sides of the inequality.
You get 4x < 150
Now divide both sides by 4.
You get x < 37.5
So x can be anything less than 37.5, such as 36, 35 or 34.

8 0

3 years ago

find the number of different outfits that can be made if you choose 1 each from 11 skirts, 9 blouses, 3 belts, and 7 pair of sho

zhenek [66]

Counting principle
Multiply 11 * 9 * 3 * 7 = 2079

6 0

3 years ago