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lora16 [44]
3 years ago
8

Regular price $4 discount 15%

Mathematics
1 answer:
Allushta [10]3 years ago
7 0
The answer would be $3.40
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\lim _{x\to 0}\left(\frac{2x\ln \left(1+3x\right)+\sin \left(x\right)\tan \left(3x\right)-2x^3}{1-\cos \left(3x\right)}\right)
Vinvika [58]

\displaystyle \lim_{x\to 0}\left(\frac{2x\ln \left(1+3x\right)+\sin \left(x\right)\tan \left(3x\right)-2x^3}{1-\cos \left(3x\right)}\right)

Both the numerator and denominator approach 0, so this is a candidate for applying L'Hopital's rule. Doing so gives

\displaystyle \lim_{x\to 0}\left(2\ln(1+3x)+\dfrac{6x}{1+3x}+\cos(x)\tan(3x)+3\sin(x)\sec^2(x)-6x^2}{3\sin(3x)}\right)

This again gives an indeterminate form 0/0, but no need to use L'Hopital's rule again just yet. Split up the limit as

\displaystyle \lim_{x\to0}\frac{2\ln(1+3x)}{3\sin(3x)} + \lim_{x\to0}\frac{6x}{3(1+3x)\sin(3x)} \\\\ + \lim_{x\to0}\frac{\cos(x)\tan(3x)}{3\sin(3x)} + \lim_{x\to0}\frac{3\sin(x)\sec^2(x)}{3\sin(3x)} \\\\ - \lim_{x\to0}\frac{6x^2}{3\sin(3x)}

Now recall two well-known limits:

\displaystyle \lim_{x\to0}\frac{\sin(ax)}{ax}=1\text{ if }a\neq0 \\\\ \lim_{x\to0}\frac{\ln(1+ax)}{ax}=1\text{ if }a\neq0

Compute each remaining limit:

\displaystyle \lim_{x\to0}\frac{2\ln(1+3x)}{3\sin(3x)} = \frac23 \times \lim_{x\to0}\frac{\ln(1+3x)}{3x} \times \lim_{x\to0}\frac{3x}{\sin(3x)} = \frac23

\displaystyle \lim_{x\to0}\frac{6x}{3(1+3x)\sin(3x)} = \frac23 \times \lim_{x\to0}\frac{3x}{\sin(3x)} \times \lim_{x\to0}\frac{1}{1+3x} = \frac23

\displaystyle \lim_{x\to0}\frac{\cos(x)\tan(3x)}{3\sin(3x)} = \frac13 \times \lim_{x\to0}\frac{\cos(x)}{\cos(3x)} = \frac13

\displaystyle \lim_{x\to0}\frac{3\sin(x)\sec^2(x)}{3\sin(3x)} = \frac13 \times \lim_{x\to0}\frac{\sin(x)}x \times \lim_{x\to0}\frac{3x}{\sin(3x)} \times \lim_{x\to0}\sec^2(x) = \frac13

\displaystyle \lim_{x\to0}\frac{6x^2}{3\sin(3x)} = \frac23 \times \lim_{x\to0}x \times \lim_{x\to0}\frac{3x}{\sin(3x)} \times \lim_{x\to0}x = 0

So, the original limit has a value of

2/3 + 2/3 + 1/3 + 1/3 - 0 = 2

6 0
3 years ago
Help me please i need to finish this today
Yakvenalex [24]

Answer: I think it is B

Step-by-step explanation:

7 0
3 years ago
Which function has no horizontal asymptote
Nookie1986 [14]
If the polynomial in the numerator is a higher degree than the denominator , there is no horizontal asymptote
7 0
3 years ago
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The ratio of the number of galleons to the number of galleys in the spanish armada is 5 : 1 . what missing information could be
Temka [501]

The amount of the galleys or the number of galleys.

If given the amount of galleons, you can find the amount of galleys by dividing the number of given galleons by 5.

If given the amount of galleys, you can find the amount of galleons by multiplying the number of given galleys by 5.

Learn more about Missing Ratio on:

brainly.com/question/16711686

#SPJ4

5 0
2 years ago
Plss solve this problem... will mark brainiest
Sonja [21]

Answer:

87.97 or 88

Step-by-step explanation:

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