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nikitadnepr [17]
3 years ago
6

Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it. lim x→[infinity]

ln(5x) 5x Step 1 As x → [infinity], ln(5x) → and 5x → .
Mathematics
1 answer:
Kobotan [32]3 years ago
6 0

Answer:

\lim_{x \to \infty} \frac{ln(5x)}{5x} = \lim_{x \to \infty} \frac{1/x}{5} = \lim_{x \to \infty} \frac{1}{5x} = 0

Step-by-step explanation:

L'Hopital's rule says that, if both numerator and denominator diverge, then we can look at the limit of the derivates.

Here we have:

\lim_{x \to \infty} \frac{ln(5x)}{5x}

The numerator is ln(5x) and when x tends to infinity, this goes to infinity

the denominator is 5x, and when x tends to infinity, this goes to inifinity

So both numerator and denominator diverge to infinity when x tends to infinity.

Then we can use L'Hopithal's rule.

The numerator is:

f(x) = Ln(5x)

then:

f'(x) = df(x)/dx = 1/x

and the denominator is:

g(x) = 5*x

then:

g'(x) = 5

So, if we use L'Hopithal's rule we get:

\lim_{x \to \infty} \frac{ln(5x)}{5x} = \lim_{x \to \infty} \frac{1/x}{5} = \lim_{x \to \infty} \frac{1}{5x} = 0

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Define four symbolic constants that represent integer 25 in decimal, binary, octal, and hexadecimal formats
stealth61 [152]

Answer:

1) Decimal    25= (25)_{10}

2) Binary    25= (11001)_2

3) Octal   25= (31)_8

4) Hexadecimal   25= (19)_{16}

Step-by-step explanation:

Given : Integer is 25                      

To find : Represent integer in decimal, binary, octal, and hexadecimal formats.

Solution :

1) Integer into decimal - To convert into decimal the base goes to 10.

So, 25= (25)_{10}

2) Integer into binary - To convert into binary the base goes to 2, it form in 0 and 1 and we divide integer by 2.

Divide 25 by 2 and note down the remainders.

2 | 25

2 | 12    R=1    ←

2 | 6     R=0   ↑

2 | 3      R=0  ↑

2 | 1  →   R=1   ↑

So,   25= (11001)_2

3) Integer into octal - To convert into octal the base goes to 8 and we divide integer by 8.

Divide 25 by 8 and note down the remainders.

8 | 25

  | 3 →   R=1    

So,   25= (31)_8

4) Integer into hexadecimal - To convert into hexadecimal the base goes to 16 and we divide integer by 16.

Divide 25 by 16 and note down the remainders.

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2 years ago
A 25​-foot ladder is placed against a vertical wall. Suppose the bottom of the ladder slides away from the wall at a constant ra
dsp73

Answer: The ladder is sliding down the wall at a rate of 5\dfrac{17}{50}\ ft/sec

Step-by-step explanation:

Since we have given that

Length of ladder = 25 foot

Distance from the wall to the bottom of ladder = 15 feet

Let base be 'x'.

Let length of wall be 'y'.

So, by pythagorus theorem, we get that

x^2+y^2=25^2\\\\15^2+y^2+625\\\\225+y^2=625\\\\y^2=625-225\\\\y^2=400\\\\y=\sqrt{400}\\\\y=20\ feet

\dfrac{dy}{dx}=-4\ ft/sec

Now, the equation would be

x^2+y^2=625\\

Differentiating w.r.t x, we get that

2x\dfrac{dx}{dt}+2y.\dfrac{dy}{dt}=0\\\\2\times 15\dfrac{dx}{dt}+2\times 20\times -4=0\\\\30\dfrac{dx}{dt}-160=0\\\\\dfrac{dx}{dt}=\dfrac{160}{30}=5.34\ ft/sec

Hence, the ladder is sliding down the wall at a rate of 5\dfrac{17}{50}\ ft/sec

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