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I was seeing an example on a book which says that: $$\frac{t}{t(x+t+1)} = \frac{1}{x+t+1}$$

The instructions read: "combine products of like terms"

What exactly is that? Why does it work and why didn't he just cut out the $t$ on the numerator and the one multiplying the denominator? He could do that, right?

Delta
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If $t \neq 0$ you could just divide the denominator and numerator by $t$, if not you need to distinguish a special case where $t=0$. By the edits I see I will edit this too. Because we take the limit $t\rightarrow 0$, $t$ will never be $0$ but we can take it as close to zero as we like, so we can divide the numerator and the denominator by $t$, or factor out in the fashion as suggested.

Rick Decker
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user112167
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Yes; I would describe this just cancelling $t$ in the numerator and denominator. I have no idea why the writer would consider this as “combining like terms”.

(If $t = 0$, then of course you can’t cancel $t$. But in that case, the left-hand fraction is undefined in any case.)

Peter LeFanu Lumsdaine
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    The writer may have been clumsily suggesting that it be viewed as $$\frac{t}{t(x+t+1)}=\frac{t\cdot 1}{t(x+t+1)}=\frac{t}t\cdot\frac1{x+t+1}=\frac1{x+t+1};.$$ – Brian M. Scott Dec 06 '13 at 23:00
  • Oh, sorry, I didn't put the part where he says that $t\to0$ so assuming you can't cancel $t$, I want to know what is that thing he does in order to flip the first expression into the second one. Edit: Ninja xD – Delta Dec 06 '13 at 23:00