hectictar

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

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?

Is the following an acceptable way to do it?

\(\phantom{=\quad}\left|\left(1+2i\right)^8\right|\\~\\ {=\quad}\left(\left|1+2i\right|\right)^8\\~\\ {=\quad}\left(\sqrt{1^2+2^2}\right)^8\\~\\ {=\quad}\left(\sqrt{5}\right)^8\\~\\ {=\quad}5^{\frac82}\\~\\ {=\quad}5^4\\~\\ {=\quad}625\)_

Thank you Melody!

I just found this:

The left-to-right rule does work for evaluating these expressions. However, in the case of division, instead of thinking that operations on the same level are done "left to right," I prefer to think that only the "item" immediately following the division symbol goes in the denominator, where an "item" is either a number or an expression enclosed in parenthesees.

The expression   \(1\div2+3+4\)   is equal to   \(\frac12+3+4\)   because we place

only the item immediately following the division symbol,  2 ,  in the denominator.

The expression   \(1\div(2+3)+4\)   is equal to   \(\frac{1}{(2+3)}+4\)   because we place

only the item immediately following the division symbol,  (2 + 3) ,  in the denominator.

Following this rule also means the expression   \(12/6/3\)   is equal to   \(\dfrac{\frac{12}{6}}{3}\)   which is  \(\dfrac23\)

The alternative rule would be that everything after the division symbol goes the denominator.

I personally do not like to say we must follow "left to right" because

for expressions like this:  1 + 2 + 3 + 4 + 5

we don't have to follow left to right. We could evaluate it like this:

1 + 2 + 3 + 4 + 5   =   1 + 2 + 3 + 9   =   1 + 5 + 9   =   10 + 5   =   15

It doesn't hurt to follow left-to-right to evaluate that expression; it just isn't necessary.

Another reason I prefer the rule "only use the item immediatley adjacent to the operator as the operand" is that we can use a similar rule for exponents. In the case of exponents, the rule would be "only the item immediately preceding the caret is the base."

4^2^3

Should  4^2^3  be  4^(2^3)  or  should it be  (4^2)^3   ?

If we just stick with the left-to-right rule, we would get

4^2^3  =  16^3  =  4096

If we use the rule "only the item immediately preceding the caret is the base," we would get

4^2^3  =  4^8  =  65536

Here, WolframAlpha says the answer is  65536:

Thanks EP for that link, but I'm pretty sure the information in that article is definitely incorrect.

The question is:

\(8\div2(2+2)\ =\ ?\)

And the ONLY correct answer is:

\(8\div2(2+2)\ =\ 8\div2(4)\ =\ \dfrac82(4)\ =\ 4(4)\ =\ 16\)

See here that WolframAlpha says the answer is 16:

Thank you! Glad it was helpful