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Verified Answer :

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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