CPhill

Here's (b) again

lim x → 3     (x - 3) / (1/x - 1/3)^2

Note that, getting a common denominator,   (1/x - 1/3)   becomes (3 - x) / 3x......so we have

lim x → 3     (x - 3) / [(3 - x) / 3x]^2        and notice that we can write the numerator as -(3 - x)......and this "cancels"  with one of the linear factors of (x - 3)^2 in the denominator leaving just -1 in the numerator....so we have...

lim x → 3     -1 ÷ [(3 - x) / [3x]^2].......so, if we had   -1 ÷ (5/6)    we would just invert the second fraction and change to multiplication.....doing the same thing here, we have.....

lim x → 3    -1 x   [3x]^2 / (3 - x)  =

lim x → 3    - [3x]^2 /(3 - x)

Note, that when x → 3^-, this function is negative. And when x  → 3⁺ , it's positive. Thus, the limit does not exist.

We have a slope (2) and a point (1, 0)

We could use something known as the point - slope equation to get to the slope-intercept form, but here's a faster way. Note that, if the slope is 2, this means that everytime x changes by a positive 1, y changes by a positive 2. And everytime x changes by a negative 1, y changes by a negative 2. So, when x changes from 1 to 0, it changes by a negative 1, so y changes by negative 2.... from 0 to -2. So (0, -2) is on the graph...so the y intercept is at -2.

So......the slope intercept form  is given by:

y = mx + b     where m is the slope and b is the y intercept.   So we have

y = 2x - 2     and that's slope-intercept form

To get standard form, get x and y on one side of the equation and the constant on the other

So we have   (noting that y  = 1y)

 y = 2x - 2   or  1y = 2x - 2           subtract 1y from both sides  and add 2 to both sides....so we have,,,

2 = 2x - 1y     or      2x - 1y = 2       and that's standard form

Here's another way to do (a)

lim x → 0⁺  √[1 - cos(x)] / x        multiply top and bottom by √[1 + cos(x)]   so we have

lim x → 0⁺  √[1 - cos^2(x)] /[ (x) √[1 + cos(x)]    =

lim x → 0⁺  sin(x) /[ (x) √[(1 + cos(x)]  =

lim x → 0⁺  [sin(x) / (x)] * [1/ √[1 + cos(x)]      and, using an identity...... lim x → 0⁺  [sin(x) / (x)]  = 1

So we have

lim x → 0⁺  [1/ √[1 + cos(x)]       and taking the limit, we have

[1/ √[1 + 1]  =  1 / √2

Just use the on site calculator....it will give you the decimal equivalent....

23% = .23      "of" means multiply.....so.....

.23 x $20 = $4.60

We have 20 colors and we want to choose 3 of them....so we have

C(20, 3) = 1140 combinations

What do you want to do with this??? Graph it??

We can choose any of the 8  for first place

Then, we can choose any of the remaining 7 for second place

Then, we can choose any of the remaining 6 for third place

So  8 x 7 x 6  = 336 ways

x 275 310 281 250 195 325 292 182 271 263 

y 76   77   73   72   70   78  75   72   75   73

Just pair up the data in order....thus...we would have the points

(275, 76)  (310, 77) (281, 73)......etc........

Then.....just plot these on a suitable graph.....

Does that help???

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