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