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3 years ago

Find the exact value of the following limit: lim x approaches 0 e^6x... check photo above.

Mathematics

1 answer:

horrorfan [7]3 years ago

8 0

Substituting <em>x</em> = 0 directly gives the indeterminate form 0/0, so you can use l'Hopital's rule (and you'll have to do that twice):

$\displaystyle\lim_{x\to0}\frac{e^{6x}-6x-1}{x^2}=\lim_{x\to0}\frac{6e^{6x}-6}{2x}=\lim_{x\to0}\frac{36e^{6x}}2$

The limand is continuous everywhere, so you can plug in <em>x</em> = 0 to get 36•1/2 = 18.

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A business has two loans totaling $50,000. One loan has a rate of 8% and the other has a rate of 12%. This year, the business ex

Debora [2.8K]

Answer:

the 8% loan has a principal of $37500

the 12% loan has a principal of $12500

Step-by-step explanation:

Let's start by writing the general equation for the interest hwre I is the interest, P is the principal (in our case would be loan amounts), "r" is the interest rate in decimal form (in our case one would be 0.12, and the other one 0.08), and t is the time in years (in our case 1 year).

$I=P*r*t$

Then we write the interest equation coming from each loan at the end of this year (we call I1 the interest coming from the 12% loan and I2 the interest coming from the 8% one). Since we don't know the loan amounts (in fact those are what we need to find) we will name one "x" and the other "y":

$I=P*r*t\\I1=x * 0.12*1\\I2=y*0.08*1$

Next, we add these last two equations term by term, and replace the addition of both interests by $4500 as given in the information:

$I1=x * 0.12*1\\I2=y*0.08*1\\I1+I2 = 0.12x+0.08y\\4500=0.12x+0.08y$

This is our first equation in the variables x and y which are our unknowns.

Now we generate the second equation on x and y by writing in agebraic terms the other piece of information we have: "the total of the two loans is $50000. That is the addition of the principals x and y should equal $50000:

$x+y=50000$

We solve for y in this last equation and replace its form in terms of x in the equation of the interest, and solve for the unknown x:

$y=50000-x\\4500 = 0.12x +0.08 y\\4500=0.12x+0.08(50000-x)\\4500=0.12x+4000-0.08x\\4500=0.12x-0.08x+4000\\4500=0.04x+4000\\4500-4000=0.04x\\500=0.04x\\x=\frac{500}{0.04} =12500$

Therefore the amount of the loan at 12% is $12500

Now to find the amount of the second loan "y" we use the equation for the totals of the loans:

$x+y=50000\\12500+y=50000\\y=50000-12500=37500$

Therefore, the loan at 8% is $37500

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3 years ago

How many times greater is the value of the 2 in 204,936 than the value of the 2 in 124,936

Sliva [168]

I believe it would be C.10

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Center c of the circle above has coordinates of (4, 3). what is the circumference of the circle?

Angelina_Jolie [31]

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3 years ago

A farmer grows two types of grain in his fields. He has a seed budget of $17,000.00 and has spent $9,200.00 to plant wheat seed

Zarrin [17]

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3 years ago

If the two legs of an isosceles right triangle measure 7 in which of the following represents the length of its hypotenuse

anyanavicka [17]

Answer:

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Step-by-step explanation:

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Find the exact value of the following limit: lim x approaches 0 e^6x... check photo above.​

Find the exact value of the following limit: lim x approaches 0 e^6x... check photo above.