How Does $\frac{1+\sin(x)}{x}=1 + \frac{\sin(x)}{x}$? I am asking this because I have faced a (limits) question where, in order to solve the equation, I had to do the following steps: 

I am not sure how the second step went from having a denominator of  $\frac{1+\sin(x)}{x}$ to having a denominator of $1 + \frac{\sin(x)}{x}$ in the third step. Can anyone please help explain how this is possible, to me?
All help is greatly appreciated.

There are two cancelling errors in that graphic: from the first step to the second, and the second to the third. You saw only the second error. The sequence of equations should begin
$$\lim_{x\to 0}\frac{\sin(x)}{x+\sin(x)}
=\lim_{x\to 0}\frac{\frac{\sin(x)}x}{\frac{x+\sin(x)}x}
=\lim_{x\to 0}\frac{\frac{\sin(x)}x}{1+\frac{\sin(x)}x}$$
The probable cause is a typographic error in the second step, typing a $1$ rather than an $x$.
