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1.

\(\lim \limits_{x \to 0}\left(\dfrac{\dfrac{1}{\sqrt{1+x}}-1}{x}\right) \)

\(\begin{array}{|rcll|} \hline && \mathbf{\lim \limits_{x \to 0}\left(\dfrac{\dfrac{1}{\sqrt{1+x}}-1}{x}\right) } \\\\ &=&\lim \limits_{x \to 0}\left(\dfrac{ \left(1+x \right)^{-\frac{1}{2}} -1}{x}\right) \quad | \quad \text{L'Hospital's rule} \\\\ &=&\lim \limits_{x \to 0} \left( \dfrac{ \dfrac{ d\ \left( \left(1+x \right)^{-\frac{1}{2}} -1 \right) } { dx } } { \dfrac{d\ (x)}{dx} } \right) \\\\ &=&\lim \limits_{x \to 0} \left( \dfrac{ -\frac{1}{2} \left(1+x \right)^{-\frac{1}{2}-1} } {1} \right) \quad | \quad x\to 0 \\\\ &=& \mathbf{-\dfrac{1}{2}} \\ \hline \end{array}\)

Find the projection of \(\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\) onto the plane \(3x - y + 4z = 0\).

\(\text{Let $\vec{P}= \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$}\)

1.

Normal vector \(\mathbf{\vec{n}}\)

\(\begin{array}{|rcll|} \hline \vec{n} &=& \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix} \quad | \quad \text{plane: }{\color{red}3}x {\color{red}-1}y + {\color{red}4}z = 0 \\ \hline \end{array}\) 

2. \(\mathbf{\vec{P}_{Proj.}}\)

\(\begin{array}{|rcll|} \hline \mathbf{\vec{P}_{Proj.}} &=& \mathbf{\vec{P}+\lambda\vec{n}} \\\\ \vec{P}_{Proj.} &=& \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}+\lambda\begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix} \\\\ x_{Proj.} &=& 1+3\lambda \\ y_{Proj.} &=& 2-\lambda \\ z_{Proj.} &=& 3+4\lambda \\ \hline \end{array} \)

3. \(\mathbf{\lambda}\)

\(\begin{array}{|rcll|} \hline 3x_{Proj.} - y_{Proj.} + 4z_{Proj.} &=& 0 \\ 3(1+3\lambda) - (2-\lambda) + 4(3+4\lambda) &=& 0 \\ 3+9\lambda-2+\lambda+12+16\lambda &=& 0 \\ 26\lambda &=& -13 \\\\ \lambda &=& -\dfrac{13}{26} \\\\ \mathbf{\lambda} &=& -\mathbf{\dfrac{1}{2}} \\ \hline \end{array}\)

4.
\(\mathbf{\vec{P}_{Proj.}}\)

\(\begin{array}{|rcll|} \hline \vec{P}_{Proj.} &=& \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}+\lambda\begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix} \quad | \quad \lambda = -\dfrac{1}{2} \\\\ &=& \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} -\dfrac{1}{2}\begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix} \\\\ &=& \begin{pmatrix} 1-\dfrac{3}{2} \\\\ 2+\dfrac{1}{2} \\\\ 3-\dfrac{4}{2} \end{pmatrix} \\\\ \mathbf{\vec{P}_{Proj.}}&=& \begin{pmatrix} \mathbf{-\dfrac{1}{2}} \\\\ \mathbf{\dfrac{5}{2}} \\\\ \mathbf{1} \end{pmatrix} \\ \hline \end{array}\)

\(|2x+2|\geq 2\\~\\ 2x+2\geq 2 \Rightarrow x\geq 1\\ 2x+2 \leq -2 \Rightarrow x \leq -2\\~\\ S = \{x : x \in (-\infty, -2] \cup [1,\infty)\}\)

I leave the second one to you.

the website has been here since 2006

its a dead rapper, that turned into a meme :') 

LOL Hectictar is right.

A square has every property that a rectandle must has (plus more)  therefore it is a rectangle.

It is like this:

A terrier  is a dog.     It has every requirement to be a dog   (but it has extra requirements as well.)

Therefore, although all terriers are dogs, not all dogs are terriers.

All squares are rectangles but not all rectangles are squares.

\(\text{Notice the expression is symmetrical in $x$ and $y$}\\ \text{This suggests that any solution will be such that $x=y$}\\ $\text{Letting $x=y$ we have}\\ 3x^4 - 17x^2 + 20 = 0\\ \text{let $u=x^2$}\\ 3u^2 - 17u + 20 = 0\\ \text{Apply the quadratic formula}\\ u = \dfrac{17 \pm \sqrt{17^2 - (4)(3)(20)}}{6} = \left(\dfrac 5 3,~4\right)\\ \text{you want integer solutions so we can ignore the first one}\\ x^2 = 4\\ x=\pm 2\)

\(\text{So the max of $x+y$ is $2+2=4$}\)

Here's a pic:

If  PQ = 12 ,  then the area of              is \(\mathcal A\)

Is something supposed to go in the blank?

Euclid gave us a definition of a "rectangular parallelogram" – a "right-angled parallelogram" which we now call a "rectangle" – that does not exclude squares, and this was written around 300 BC (much longer than 55 years ago!).

Math has improved and developed over time, guest. There are new concepts being taught in school. As our civilization and technology improves, our school must keep up with it.

One mathematical definition of a rectangle is  "a quadrilateral with four right angles."

Never heard of it. When I went to school, some 55 years ago, a square was a square, and a rectangle was a rectangle.

We can use a semicolon to join to otherwise complete sentences. So

"The setting makes you more likely to anticipate the ending, as this is in a small rural village and anything can happen; although it doesn’t completely reveal the ending and still preserves the surprise."

is completely grammatically equivalent to:

"The setting makes you more likely to anticipate the ending, as this is in a small rural village and anything can happen. Although it doesn’t completely reveal the ending and still preserves the surprise."

Let's look at it the second way for now and convert it back later.

Now another problem is more apparent. "Although" is a subordinating conjunction. Saying "Although it doesn't completely reveal the ending and still preserves the surprise." is like saying "After he went home." In both cases, the reader is left hanging, waiting for the rest of the sentence. "After he went home" -- then what happened?

Here's a short but helpful post about it:

a  and  b  do not have to be different in order for it to be a rectangle. A square is a rectangle.

1.

\(\phantom{=\quad}\lim\limits_{x\to0}\left(\dfrac{\frac{1}{\sqrt{1+x}}-1}{x}\right)\\~\\ {=\quad}\lim\limits_{x\to0}\left(\dfrac{\frac{1}{\sqrt{1+x}}-\frac{\sqrt{1+x}}{\sqrt{1+x}}}{x}\right)\\~\\ {=\quad}\lim\limits_{x\to0}\left(\dfrac{\frac{1-\sqrt{1+x}}{\sqrt{1+x}}}{x}\right)\\~\\ {=\quad}\lim\limits_{x\to0}\left({\frac{1-\sqrt{1+x}}{\sqrt{1+x}}}\div{x}\right)\\~\\~\\ {=\quad}\lim\limits_{x\to0}\left({\frac{1-\sqrt{1+x}}{\sqrt{1+x}}}\cdot\frac{1}{x}\right)\\~\\~\\ {=\quad}\lim\limits_{x\to0}\left({\frac{1-\sqrt{1+x}}{x\sqrt{1+x}}}\right)\)

When you get to this point, instead of multiplying the numerator and denominator by   \(\sqrt{1+x}\) ,

try multiplying the numerator and denominator by   \(1+\sqrt{1+x}\)   (the conjugate of the numerator)

After you do that, you should be able to simplify it to a point where you can substitute  0  in for  x  and get a defined value.

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Hope this helps for your first question!

As for your second question, have you learned about or are you allowed to use L'Hopital's rule?

idk ...maybe the forum doesnt like you that much yet......

better post crazy math questions...

thank you

The website back a few months was very different than it is now. Beside the +1 thing , you see a heart, that heart wasnt there, it was just the +1 and the more if those you have , your name appears on the list of users on the right side. When i first joined , i wanted to get more points than Melody, but that never really happened, you cant beat someone as intelligent as her, i mean im just a teenager anyway so.....

i joined in 2013 and Melody and CPhill has joined way earlier than that so this website is pretty old....

The points you get are basically just earnings kind of this, they also have a competitive element to them , so if you have  A LOT of time and INTELLIGENCE you can try to score as many points as possible.

I remember being so crazy for these things that one when the site was being updated, I lost my points , a lot of them, and I got so worried, then someone helped me out by giving +1 on my old posts.

i dont remember what we called them back then but i guess it was just 'score' or '+1s'....

So you round 403 to the nearest hundred, getting 400.

Then to find the total cost you multiply 400 by 60. Because there are 12 months in a year for 5 years, so 5 * 12 = 60.

You get 24000

Ok that is your estimate

Then the calculator time.

403 * 60 = 24180 

You are 180 off.

Now i let u answer the question: Were you reasonable in your estimate?

The answer provided by CalculatorUser is pretty much correct , but i would like to add some corrections according to myself.

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Subject:

The subject of the letter was to tell the clergymen his reason for why he had come to Birmingham. In the letter , he stated that he was here to achieve equal rights for the people whom were oppressed. He also stated that he was going to use nonviolent means such as protesting and negotiating to achieve his goal of equality.

Occasion:

Martin Luther King Jr. wrote this letter inside Birmingham jail because he was accused of his improper acts of civil disobedience. Dr. King was doing using these protesting techniques in order to express his disappointment at the people and laws who which were suppressing the black community. He wanted to fix the unjust actions that were implemented into in the society. 

Audience:

Martin Luther King Jr. wrote the Letter from Birmingham jail as a response to eight clergymen’s statement in the local newspaper urging black people to withdraw their support from him and his demonstrations. 

Purpose:

The purpose of writing the letter was to respond to the publishing in the local newspaper thom that was posted by eight clergymen in Alabama. The newspaper that the clergymen posted was asking Black’s black people  to not change their current way of life, but to stay oppressed and submissive to the white man men. Dr. King’s response to the statement was that without any action , no result would occur , and so he wants them to take a stance without violence , as he believes that would only escalates the situation. It also explains his reasons for starting the civil disobedience. He wanted to change the way that people were  treated all over the country and explains the horrible way that African Americans are were being treated. 

Speaker

Martin Luther King Jr. is the speaker in the letter. He was a civil rights activist for many years and wanted to see all the people being treated equally. 

Tone

The tone that is conveyed in the letter shifts and changes in order to match the words he was conveying at the time. He keeps a courageous and strong will throughout the speech and includes a dedication in areas such as his willingness to suffer for equality among people. He never changed the tone of the letter to anything of violence or hate. He always keeps a passive stance, a very peaceful one , in order to achieve his ultimate goal of equality among men in the country. The tone of the letter has also been changed to that of previous speeches and addresses. In  , which this he writes  wrote this to convey it to his audience which was who were primarily the reverends and pastors. Due to the  limited religious practices , he includes many religious ideals in the letter to address these men personally. 

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i tried my best , checking and rechecking it , i hope it helps to some extent.

I think you should ask your teacher about the 'tense' to use, for eg , in the 'purpose' para , past tense is used pretty much and in the 'tone' present tense  is used , so it would help you to edit better if you do that. That way you wouldnt have to swing between deciding which tense should be applied where.

I agree that there could be more rectangles; that's why I used a phrase 'so far'

Goodbye, square! Hello, rectangle!

The ratio of cups to quarts is \(4:1\); there are 4 cups in a quart. Once you know this ratio, use dimensional analysis to figure out how many quarts of mayonnaise there are. 

\(\frac{25\text{cups}}{1}*\frac{1\text{quart}}{4\text{cups}}=\frac{25}{4}\text{quarts or }6.25\text{ quarts of mayo}\)