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DIA [1.3K]
1 year ago
5

Find where the sequence converges

Mathematics
2 answers:
soldier1979 [14.2K]1 year ago
8 0

We can also apply l'Hôpital's rule by first rewriting the limit as

\displaystyle \lim_{k\to\infty} \left(1 + \frac4k\right)^k = \lim_{k\to\infty} \exp\left(\ln \left(1 + \frac4k\right)^k\right) = \exp\left(\lim_{k\to\infty} \frac{\ln\left(1+\frac4k\right)}{\frac1k}\right)

Applying the rule gives

\displaystyle \lim_{k\to\infty} \frac{\ln\left(1+\frac4k\right)}{\frac1k} = \lim_{k\to\infty} \frac{\left(-\frac4{k^2}\right)/\left(1+\frac4k\right)}{-\frac1{k^2}} = 4 \lim_{k\to\infty} \frac1{1 + 4k} = 4

so that the overall limit is

\displaystyle \lim_{k\to\infty} \left(1 + \frac4k\right)^k = \lim_{k\to\infty} \exp(4) = \boxed{e^4}

avanturin [10]1 year ago
3 0

Answer:  \\ \lim\limits_{k \to \infty} (1+\frac{4}{k})^k =e^4.

Step-by-step explanation:

\displaystyle\\ \lim_{k \to \infty} (1+\frac{4}{k})^k  \\x=\frac{x}{4} *4\\So,\  \lim_{k \to \infty} (1+\frac{4}{k})^\frac{k}{4}*4 \\ \lim_{k \to \infty} ((1+\frac{4}{k})^\frac{k}{4} )^4.\\Use\  the\  second\  wonderful\  limit:\\\boxed { \lim_{x \to \infty} (1+\frac{1}{x})^x=e  },\\\\So,\\ \lim_{k \to \infty} (1+\frac{4}{k})^k =e^4.

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jolli1 [7]

Answer:

373

Step-by-step explanation:


5 0
3 years ago
Form a polynomial f(x) with real coefficients having the given degree and zeros.
dimaraw [331]
There are many polynomials that fit the bill,
f(x)=a(x-r1)(x-r2)(x-r3)(x-r4)  where a is any real number not equal to zero.
A simple one is when a=1.
where r1,r2,r3,r4 are the roots of the 4th degree polynomial.
Also note that for a polynomial with *real* coefficients, complex roots *always* come in conjugages, i.e. in the form a&pm;bi  [&pm;=+/-]

So a polynomial would be:
f(x)=(x-(-4-5i))(x-(-4+5i))(x--2)(x--2)
or, simplifying
f(x)=(x+4+5i)(x+4-5i)(x+2)^2
=x^4+12x^3+77x^2+196x+164   [if you decide to expand]
6 0
3 years ago
50 POINTS!!! In rectangle ABCD, AB = 6 cm, BC = 8 cm, and DE = DF. The area of triangle DEF is one-fourth the area of rectangle
aalyn [17]

Answer:

EF=4\sqrt{3}

Step-by-step explanation:

In rectangle ABCD, AB = 6, BC = 8, and DE = DF.

ΔDEF is one-fourth the area of rectangle ABCD.

We want to determine the length of EF.

First, we can find the area of the rectangle. Since the length AB and width BC measures 6 by 8, the area of the rectangle is:

A_{\text{rect}}=8(6)=48\text{ cm}^2

The area of the triangle is 1/4 of this. Therefore:

\displaystyle A_{\text{tri}}=\frac{1}{4}(48)=12\text{ cm}^2

The area of a triangle is half of its base times its height. The base and height of the triangle is DE and DF. Therefore:

\displaystyle 12=\frac{1}{2}(DE)(DF)

Since DE = DF:

24=DF^2

Thus:

DF=\sqrt{24}=\sqrt{4\cdot 6}=2\sqrt{6}=DE

Since ABCD is a rectangle, ∠D is a right angle. Then by the Pythagorean Theorem:

(DE)^2+(DF)^2=(EF)^2

Therefore:

(2\sqrt6)^2+(2\sqrt6)^2=EF^2

Square:

24+24=EF^2

Add:

EF^2=48

And finally, we can take the square root of both sides:

EF=\sqrt{48}=\sqrt{16\cdot 3}=4\sqrt{3}

6 0
3 years ago
Read 2 more answers
In a recipe, the ratio of fluid ounces of water to fluid ounces of tomato paste is 3:4. You plan to make 35 fluid ounces of sauc
WITCHER [35]
3:4......added = 7

3/7(35) = 105/7 = 15 oz water
4/7(35) = 140/7 = 20 oz tomato paste <==
4 0
3 years ago
3.
Zanzabum

Answer:

see below

Step-by-step explanation:

y = 8x − 9

y = 4x − 1

Since both equations are equal to y, we can set them equal to each other

8x-9 = 4x-1

Subtract 4x from each side

8x-4x-9 = 4x-4x-1

4x-9 = -1

Add 9 to each side

4x-9+9 = -1+9

4x = 8

Divide each side by 4

4x/4 = 8/4

x=2

Now find y

y = 4x-1

y = 4(2)-1

y = 8-1

y = 7

(2,7)

The point of intersection when the 2 lines are graphed is (2,7)

7 0
3 years ago
Read 2 more answers
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