I am preparing for a test and the following question is from the practice exam. My problem is that the explanation skips over things that I don't understand. I was hoping somebody here could clear it up for me.

Find the limit:

lim x(sin 5x) / (1 − cos x) = ?

x→0

I know that : x sin x / x = 1

and : 1-cos x / x^2 = 1/2

And these come into play in this problem.. .it's the "getting there" I don't understand. here's the solution given:

lim x(sin 5x) / (1 − cos x) = lim 5x^2(sin 5x) / 5x(1-cosx) = 5/(1/2)

x→0 x→0

so the answer is 10.

The solution never shows where the known things above actually appear. where is the x that x sin x is divided by? Where is the x^2 that 1-cosx is divided by?

I am sure this is supposed to be understood, but I don't understand it. Please help.