# Thread: Factoring an X out

1. ## Factoring an X out

I am stuck on the algebra in a basic limit problem example.

(sqrt of x^2 + x) + x

They factor out an x and get

x * [(sqrt of 1 + 1/x) + 1]

This is just the denominator of the problem but it is the part I dont understand.

2. $\displaystyle \displaystyle \sqrt{x^2 + x} + x = \sqrt{x^2\left(1 + \frac{1}{x}\right)} + x$

$\displaystyle \displaystyle = \sqrt{x^2}\sqrt{1 + \frac{1}{x}} + x$

$\displaystyle \displaystyle = x\sqrt{1 + \frac{1}{x}} + x$

$\displaystyle \displaystyle = x\left(\sqrt{1+\frac{1}{x}} + 1\right)$.

3. Thanks. Makes sense now that I see what you did. I dont think I would have ever thought to do the first step. Guess I need more practise.

4. When doing this:
$\displaystyle \displaystyle \sqrt{x^2 + x} + x = \sqrt{x^2\left(1 + \frac{1}{x}\right)} + x$

Why does the $\displaystyle x$ turn into $\displaystyle (1+\frac{1}{x})$?

I can't see it...

5. What's $\displaystyle \displaystyle x \div x^2$?

6. $\displaystyle \frac{1}{x}$

7. Originally Posted by MaverickUK82
Why does the $\displaystyle x$ turn into $\displaystyle (1+\frac{1}{x})$?
Well of course it does not do that.
Do you see that $\displaystyle \displaystyle x^2\left(1 + \frac{1}{x}\right)=x^2+x~?$

8. Yes, I totally understand now.

Had a blank moment there and didn't fully understand (or read) the question...

9. Originally Posted by Prove It
$\displaystyle \displaystyle \sqrt{x^2 + x} + x = \sqrt{x^2\left(1 + \frac{1}{x}\right)} + x$

$\displaystyle \displaystyle = \sqrt{x^2}\sqrt{1 + \frac{1}{x}} + x$

$\displaystyle \displaystyle = x\sqrt{1 + \frac{1}{x}} + x$

$\displaystyle \displaystyle = x\left(\sqrt{1+\frac{1}{x}} + 1\right)$.
It should be noted that $\displaystyle \sqrt{x^2} = |x|$ not $\displaystyle x$. Since the OP said "I am stuck on the algebra in a basic limit problem example." this fact is likely to be important in getting the correct answer to the actual question.

10. Originally Posted by mr fantastic
It should be noted that $\displaystyle \sqrt{x^2} = |x|$ not $\displaystyle x$. Since the OP said "I am stuck on the algebra in a basic limit problem example." this fact is likely to be important in getting the correct answer to the actual question.
Thats is a good point. I always space that fact. The original question in the book was a limit a infinitey with the problem I stated in the numerator. They use the conjugate and in the end cancel out the X top and bottom and end up with an answer of 1/2. If I new Latex it would be more apparent. Dont know if the limit at infinitey has anything to do with not needing the absolute value.

11. Since you're going to a positive value (I assume you mean $\displaystyle \displaystyle \lim_{x \to +\infty}$), you should note that $\displaystyle \displaystyle |x| = x$, so the modulus signs would not be necessary in this case...

12. Originally Posted by Prove It
Since you're going to a positive value (I assume you mean $\displaystyle \displaystyle \lim_{x \to +\infty}$), you should note that $\displaystyle \displaystyle |x| = x$, so the modulus signs would not be necessary in this case...
A limit question requiring this sort of algebraic re-arrangement only makes sense if x is approaching $\displaystyle \displaystyle -\infty$. It's trivial if $\displaystyle \displaystyle x \to +\infty$ and no algebraic arrangement is necessary.

Again we suffer from the typical ambiguity that arises when the real question does not get posted by the OP.

13. Originally Posted by mr fantastic
A limit question requiring this sort of algebraic re-arrangement only makes sense if x is approaching $\displaystyle \displaystyle -\infty$.
What about $\displaystyle \displaystyle \lim _{x \to \infty } \sqrt {x^2 + x} - x~?$

14. Originally Posted by Plato
What about $\displaystyle \displaystyle \lim _{x \to \infty } \sqrt {x^2 + x} - x~?$
That was the original question. I thought it might be more cumbersome to try and write the question without latex and the limit part seemed simple enough. I just got stumped on the algebra. Thanks for everyones help.

15. Originally Posted by mr fantastic
A limit question requiring this sort of algebraic re-arrangement only makes sense if x is approaching $\displaystyle \displaystyle -\infty$. It's trivial if $\displaystyle \displaystyle x \to +\infty$ and no algebraic arrangement is necessary.

Again we suffer from the typical ambiguity that arises when the real question does not get posted by the OP.
My question was answered in the first reply and might I add very quickly which was greatly appreciated. I only put part of the question in because it was the only part I didnt understand and seemed to fit in the algebra section instead of the calculus section. Sorry for the confusion.

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