Why are we dividing by x in the solution for this limit problem?

[rohan03]

I am studying unit on limits and one of the example given to prove and established limit simply doesn't make sense.
in the given solution of the example righhand side is simply not making sense to me - please see attached document and anyone can throw some light on this will be great so i can progress.

 

[spamiam]

From the part you circled in the document, it looks like you just need to put things over a common denominator:
$$
\frac{1}{1-x} - 1 = \frac{1}{1-x} - \frac{1-x}{1-x} = \frac{1 - (1 - x)}{1-x} = \frac{1 - 1 +x}{1-x} = \frac{x}{1-x} \, .
$$
In the next step divide through by x (which is why we must have $x \neq 0$). Then use the same trick as above, but in reverse:
$$
\frac{1}{1-x} = \frac{1 - x + x}{1-x} = \frac{1-x}{1-x} + \frac{x}{1-x} = 1 + \frac{x}{1-x} \, .
$$

 

[rohan03]

that is the step I am struggling with.
so by putting things in common denominator I get x/1-x
now if I divide by x = why are we dividing 1/(1-x) and not x/(1-x)? or we are considering
x/x + x/(1-x) = 1+ x/(1-x)
and I will appreciate it if you can make further steps of part a clear as well
Many thanks

 

[spamiam]

that is the step I am struggling with.
so by putting things in common denominator I get x/1-x
now if I divide by x = why are we dividing 1/(1-x) and not x/(1-x)? or we are considering
x/x + x/(1-x) = 1+ x/(1-x)
and I will appreciate it if you can make further steps of part a clear as well
Many thanks

We are dividing $\frac{x}{1-x}$ by x: that's why we get $\frac{1}{1-x}$. I just showed why$\frac{1}{1-x} = 1 + \frac{x}{1-x}$.