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lim((sqrt(x+3)-2)/sqrt(x-1)),x->1

\(\small{ \begin{array}{rcl} \lim \limits_{x\to 1} { \left( \dfrac{ \sqrt{x+3} -2 } { \sqrt{x-1} } \right) }=\ ?\\ \end{array} }\)

\(\small{ \begin{array}{rcl} \dfrac{ \sqrt{x+3} -2 } { \sqrt{x-1} } &=& \dfrac{ \sqrt{x+3} -2 } { \sqrt{x-1} } \cdot \left( \dfrac{ \sqrt{x+3} +2 }{ \sqrt{x+3} +2 } \right) \cdot \left( \dfrac{ \sqrt{x-1} } { \sqrt{x-1} } \right)\\ &=& \dfrac{ (x+3-4 )(\sqrt{x-1}) } { ( x-1 ) \cdot (\sqrt{x+3} +2) } \qquad (\sqrt{x+3} -2)(\sqrt{x+3} +2) = (\sqrt{x+3})^2 -2^2\\ &=& \dfrac{ (x-1)(\sqrt{x-1}) } { ( x-1 ) \cdot (\sqrt{x+3} +2) } \\ &=& \dfrac{ (\sqrt{x-1}) } { (\sqrt{x+3} +2) } \\ \lim \limits_{x\to 1} { \left( \dfrac{ \sqrt{x+3} -2 } { \sqrt{x-1} } \right) }&=& \lim \limits_{x\to 1} { \left( \dfrac{ \sqrt{x-1} } { \sqrt{x+3} +2 } \right) }\\ &=& \dfrac{ \sqrt{1-1} } { \sqrt{1+3} +2 }\\ &=& \dfrac{ \sqrt{0} } { \sqrt{4} +2 }\\ &=& \dfrac{ 0 } { 2 +2 }\\ &=& \dfrac{ 0 } { 4}\\ \mathbf{ \lim \limits_{x\to 1} { \left( \dfrac{ \sqrt{x+3} -2 } { \sqrt{x-1} } \right) } } & \mathbf{=} & \mathbf{ 0 } \end{array} }\)

What about it?

Find the average rate of change of g(x)=5x2+3x4 on the interval [-1,3]

CPhill's answer is the best one but I was playing around with this

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The instantaneaous rate of change at any given point is given by the first derivative. It is the gradient of the tangent.

\(g'(x)=10x+12x^3\)

Now, The integral sign  is a sylized S and it really stands for 'sum of'

So the sum of all the gradients from -1 to 3 is given by

\(\displaystyle \int_{-1}^3\;10x+12x^3\;dx\\\\\\ = [5x^2+3x^4]_{-1}^3\\\\\\ =(45+243)-(5+3)\\\\ =280\)

Now the average of this would have to be  

\(\frac{280}{3-\;-1} = 70\)

So the average rate of change is 70

I invite other mathematicians to comment on this.  There is something I don't quite understand but I cannot explain what it is.   

It is this statement that is causing me problems

So the sum of all the gradients from -1 to 3 is given by the integral .....

≈ 3.2964861117302469

g(x)=5x²+3x⁴ on the interval [-1,3]

The average rate of change is given by :  change in y on the interval /  change in x on the interval =

[g(3)  - g(-1)] / [3 - (-1)]  =

[288 - 8 ] / [ 4 ] =

280 / 4  =

70

How many cards are your picking?

If you are picking just one card and it is a normal 52 pack of randomly placed cards then the chance of choosing an ace is 4/52  =  1/13

Let x be the number of boys  and x + 5 be the number of girls....and we have....

[x] / [x + 5] =  3 /4     cross-multiply

4x = 3[x + 5]   simplify

4x  = 3x + 15     subtract 3x from both sides

x = 15  = the number of boys

So....there are 20 girls

Solve for m:
-3 = m+16

-3 = m+16 is equivalent to m+16 = -3:
Answer: | 
| m+16 = -3

X=-19

2000^20=1.048576 X 10^66

Thanks Guest,   

-3x(-2x+3) certainly does equal  6x^2-9x 

BUT your second statement   x=0 and x=3/2   is NOT correct.

This would only be true if the question was

-3x(-2x+3)=0

-3x(-2x+3)

This can be written like this:

6x^2 - 9x  and x=0 and x=3/2

(1.01)^14=1.149474.......etc. which is the answer.

By the Law of Cosines, we have

15^2  = 10^2 + 12^2 - 2(10)(12) cos(x)

And solving this for x, we have

cos^-1 [ ( 225 - 100 - 144) / -240 ] = x = about 85.46”.......and the angle opposite this one will be the same

The other two opposite angles will measure (180 - 85.46)° = about 94.54°  each

Using the cosine inverse, we have

cos-1(.-.06)  = theta = about 126.87°.....however......this could also be a 3rd quadrant angle =

180 + (180 - 126.87)  = about 233.13°

And substituting 180 for "pi,"  we have

And cos(pi + theta)  = cos(180 + theta) = -cos(theta)  = -cos(126.87)  = 0.6     ....and....

cos(180 +233.13) =  -cos(233.13)  = 0.6

Suppose cos (θ) = -0.6.  What is the value of cos (pi + θ)?  Anyone who can write out the steps to this problem will be awesome.  It sounds like an easy question, but I am stumped.

It appears that you are using "radians" instead of "degrees".

Cos(theta)=-.6, therefore the inverse cosine of Theta is=2.2143.... + 3.141592653=5.355892653

Therefore, Cos(5.355892653)=.6, which is your answer.

Because you are eating more calories of food than you are using/burning each day.  Period.

Distance / Rate = Time

[215 /  88.5]  - [ 215/104.6]   =  about 0.37 hrs = about 22.4 min

Just subtract 32 from both sides.....but.....a better question that you might ask yourself......."What do I need to add to 32 to get 42???"

If Chris intends to buy 3 jars of salsa and each jar costs $3, then the total cost will be 3 X 3=$9

Since each jar has 16 ounces, then, 3 X 16=48 ounces of salsa altogether.

The cost per ounce is 9/48=.1875 cents per ounce.

Since each guest will eat 3 ounces, then the cost per guest will be 3 X .1875=.5625 cents per guest.

Total cost for the party of 15 will be =15 X .5625=$8.44 (rounded)

It's already a function.....!!!!!

Each ounce costs   3.00 / 16  = 18.75 cents

So,  each person eats 3 ounces and we have.....3(18.75) = 56.25 cents

(1/6) (140 - 80)  =

(1/6) (60)  =

60 / 6   =

10

1/6 (140-80)=1/6(60)=60/6=10, which is the answer.

Thanks!

I can't say for sure....but if you're  trying to evaluate something other than a base 10 or base e log in this form :  log_a b ,  you can use the change-of-base rule that says the result of this is just :

log(b) / log (a)