Note that 1 - cos(x) = 2sin²(x/2)
⇒ lim x→0 xtan(x)/ [1 - cos(x)] =
= lim x→0 xtan(x) / 2sin²(x/2) =
= lim x→0 1/2 xtan(x) / [sin²(x/2) / (x/2)² * (x/2)²]
Also, we know that lim x→0 sin(x)/x = lim x→0 tan(x)/x = 1
So the limit is :
= 1/2 lim x→0 xtan(x) / (x²/4) =
= 1/2 lim x→0 4/x² * xtan(x) =
= 2 lim x→0 tan(x)/x =
= 2.
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