CPhill

Note that we are taking the limit as x approaches 0 from the right.......

Using the Puiseux  series expansion for the first six terms of sin (sqrt(x)), we have :

lim x →  0   [sqrt(x)-x^(3/2)/6+x^(5/2)/120-x^(7/2)/5040+x^(9/2)/362880-x^(11/2)/39916800] / x

Dividing each term by x   and taking the limit as x → 0  on the last five terms will result in 0

Dividing the first term by x , we have

lim x →  0      1 / sqrt(x)

As x → 0,    the denominator gets extremely "small" in comparison to the numerator .... so the function  →  +infinity  as x  → 0 ....i.e., .......the limit does not exist

Here's the graph that confirms this  .....   https://www.desmos.com/calculator/tzfgdcaps7

Assuming 500 is the  first term, this is equivalent to  :

500  + (50)*3  - (15)*2   =

500 +  [ 150 ] - [30]  =

650  - 30  =

620

log₃ 81 ^(-.2)  =

-.2 * log₃ 81  =

-1/5 * 4   =

-4 / 5

4^h = 1/64       note that 4³ = 64   and  1/ 4³  =  4^-3

So

4^h = 4^-3

And   h = -3

Remember that   sqrt( -1)   = i   ....so we have

sqrt(-28)  =

sqrt (-1 * 4 * 7)  =

sqrt (-1) * sqrt(4) * sqrt(7)  =

i * 2 * sqrt(7)   or    just     

2*sqrt(7) i

-4b-(-8)=24  simplify

-4b + 8 = 24     subtract 8 from both sides

-4b = 16     divide both sides by -4

b = -4

We need to copmare these two things :

1000(1 + .055/12)¹²   ≈  $ 1056.41

And

1000(1 + .054/365)³⁶⁵  = $1055.48

Looks like the monthly rate is slightly better....

Following Melody's logic that, from a point outside a circle, two tangents drawn to the circle will have equal lengths.....look at the following diagram of the problem.....

Angle OCR will ≈  tan^-1(1/2) ≈ 26.565051177078°

And we can imagine kite PORC  with angle PCR = 2* 26.565051177078°  ≈ 53.130102354156°

And the tangent of this angle will   = 4/3

So.......the line with the equation  y = 4/3x  will intersect this circle at  (4.8, 6.4)

Angle OBR ≈ tan^-1(2/3) ≈ 33.69006752598°

And, like before, we can imagine kite QORB with angle QBR = 2 * 33.69006752598° ≈ 67.38013505196°

And the tangent of this angle = 12/5

So....the line with the equation   y = (-12/5)(x - 14)  will intersect the circle at  about ( 152/13 , 72/13)

And these two tangent lines will intersect at (9, 12)  = "A"

So ....  AQ = sqrt [(9 - 152/13)^2 + ( 12 - 72/13)^2 ]  = 7

And  since AP  is a tangent to the circle drawn from "A,"  it will have the same length as AQ   = 7

Looks like you were correct, Melody.....

5/10 = 1/2

6/9  = 2/3

3/8

2/6   = 1/3

Only 3/8 cannot be reduced any further

"Of" means multiply....so we have

(1 / 16)  * 28  =

(1 / 16) *  (28 / 1)   =

28  / 16  =

7 / 4