Evaluate the limit as x approaches 0 of (1 - x^(sin(x)))/(xlog(x))aka: $(1-x^(sin(x)))/(xlog(x))$

Please include steps/explanation.

Answers

Answer 1

Answer: $sin~ x \approx x ~ ~\sf{as}~~ x \rightarrow 0$

We can replace sin x with x anywhere in the limit as long as x approaches 0.

Also,

$\large \lim_( x \to 0 ) ~ x^x = 1$

I will make the assumption that log(x)=ln(x).

The limit result can be proven if the base of log(x) is 10.

$\large \lim_(x \to 0^(+)) (1- x^(\sin x) )/(x \log x ) \n~\n \large = \lim_(x \to 0^(+)) (1- x^(\sin x) )/( \log( x^x) ) \n~\n \large = \lim_(x \to 0^(+)) (1- x^(x) )/( \log( x^x) ) ~~ \normalsize{\text{ substituting x for sin x } } \n~\n \large = (\lim_(x \to 0^(+)) (1) - \lim_(x \to 0^(+)) \left( x^(x)\right) )/( \log( \lim_(x \to 0^(+))x^x) ) = (1-1)/(\log(1)) = (0)/(0)$

We get the indeterminate form 0/0, so we have to use Lhopitals rule

$\large \lim_(x \to 0^(+)) (1- x^(x) )/( \log( x^x) ) =_(LH) \lim_(x \to 0^(+)) (0 -x^x( 1 + \log (x)) )/(1 + \log (x) ) \n = \large \lim_(x \to 0^(+)) (-x^x) = \large - \lim_(x \to 0^(+)) (x^x) = -1$

Therefore,

$\large \lim_(x \to 0^(+)) (1- x^(\sin x) )/(x \log x ) =\boxed{ -1}$

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Looking for some furniture for her new apartment, Susan visits the local swap meet in search of something cheap. Knowing that most swap meet vendors willnot accept credit cards and that she will not have any money until her next payday, Susan decides to take out a $200 cash advance on her credit card at an interest rate of 32%. If Susan had no previous balance on her credit card, and she manages to pay off the balance within 1 month, how much will she have to pay in interest?

Answers

Given:
cash advance = 200
interest rate = 32% , I'm assuming this is a per annum rate.
pay w/in a month.

I = 200 * 32% * 1/12
I = 5.33

She will have to pay 5.33 in interest after 1 month.

URGENT PLZ HELP: Graph y = tanx for -pi/4 ≤ x ≤ x/4.
What is the range?

Answers

I believe that the answer is D) all real numbers.

Find the median for the following set of numbers. 5, 19, 2, 28, 25 2 5 19 25

Answers

Step 1: List the numbers in order (Least - Greatest): 2, 2, 5, 5, 19, 19, 25, 25, 28
Step 2: Find the middle number: 2, 2, 5, 5, 19, 19, 25, 25, 28
Answer: 19
(Information for further questions): If there are 2 middle numbers, average them. (Add them up and divide by 2)

Given the information below, QRS is UVT similar to ? If so, give the scale factor.QR= 8 RS = 16 UV = 18 VT= 34 M<S=36 M<T=36

A.No, the triangles are not similar.
B.Yes, the scale factor is 1:2.
C.Yes, the scale factor is 4:9.
D.Yes, the scale factor is 8:17.

Answers

$Look\ at\ the\ picture.\n\n\Delta QRS\sim\Delta UVT\ if\ (|QR|)/(|UV|)=(|RS|)/(|VT|)\n\n(|QR|)/(|UV|)=(8)/(18)=(4)/(9)\n\n(|RS|)/(|VT|)=(16)/(34)=(8)/(17)\n\nAnswer:A.\ No,\ the\ triangles\ are\ not\ similar.$

Sam has $2.25 in quarters and dimes, and the total number of coins is 12. How many quarters and how many dimes?HELP NOW DUE In 30 min!

Answers

The solution is, Sam has 5 dimes and 7 quarters.

What is equation?

An equation is a mathematical statement that is made up of two expressions connected by an equal sign. In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal.

here, we have,

Sam has $2.25 in quarters and dimes, and the total number of coins is 12.

let, Sam has n₁ dimes and n₂ quarters.

then, we have,

n₁*$0.10 + n₂*$0.25 = $2.25

and, we have,

n₁ + n₂ = 12

This means that:

n₁ = 12 - n₂

And:

(12 - n₂)*$0.10 + n₂*$0.25 = $2.25

12*$0.10 - n₂*$0.10 + n₂*$0.25 = $2.25

$1.20 + n₂*$0.05*(5-2) = $2.25

$1.20 + n₂*$0.05*3 = $2.25

$1.20 + n₂*$0.15 = $2.25

n₂*$0.15 = $2.25 - $1.20

n₂*$0.15 = $1.05

n₂ = ($1.05)/($0.15)

n₂ = 7

If n₂ = 7:

n₁ + 7 = 12

Therefore, n₁ = 5.

Answer:

5 dimes and 7 quarters.

To learn more on equation click:

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n₁*$0.10 + n₂*$0.25 = $2.25

n₁ + n₂ = 12

-----------

This means that:

n₁ = 12 - n₂

And:

(12 - n₂)*$0.10 + n₂*$0.25 = $2.25

12*$0.10 - n₂*$0.10 + n₂*$0.25 = $2.25

$1.20 + n₂*$0.05*(5-2) = $2.25

$1.20 + n₂*$0.05*3 = $2.25

$1.20 + n₂*$0.15 = $2.25

n₂*$0.15 = $2.25 - $1.20

n₂*$0.15 = $1.05

n₂ = ($1.05)/($0.15)

n₂ = 7

If n₂ = 7:

n₁ + 7 = 12

Therefore, n₁ = 5.

---------

Answer:

5 dimes and 7 quarters.

Write an expression using the distributive property that can be used to find the quotient of 36 ÷ 3

Answers

something like 36÷3 = (3)÷3=1 and (6)÷3=3 you put 1 and 3 together to make 13

Final answer:

The solution to the given mathematical problem is found by using thedistributive property to break down 36 into its component parts (30 and 6), then dividing each part by 3, yielding the answer of 12.

Explanation:

The question is framing a scenario where we need to use the distributive property to simplify a division operation. The distributive property deals with multiplying a number by a sum or difference, in essence 'distributing' the multiplication over addition or subtraction.

So the division 36 ÷ 3 could be set up using the distributive property like (3×10) + (3×2) by breaking down 36 into a sum like 30 + 6.

So, 36 ÷ 3 = (3 x 10 + 2) ÷ 3 Applying the distributive property, we distribute the division operation over the addition. = (30 ÷ 3 + 6 ÷ 3) = 10 + 2 = 12