Evaluate the following limits : lim(x→0) (x tan x)/(1 - cos x)

Question

Evaluate the following limits : \(\lim\limits_{\text x \to0}\cfrac{\text x\,tan\,\text x}{1-cos\,\text x} \)

lim(x→0) (x tan x)/(1 - cos x)

Comments

it can be solve in more efficient way

Dividing the numerator and the denominator by x²:

 =  x tan x/ x^2
   ____________
   1-cosx/ x^2

 =  tanx
      ____
     x  
     ________
     2 sin^2 x/2  
     _________
     x/2 *x/2  *4

limit tends to 0

= 1
  _____
  2*1/4

limit tends to 0

=4/2

=2

Answer 1

\(\lim\limits_{\text x \to0}\cfrac{\text x\,tan\,\text x}{1-cos\,\text x} \)

Hence, \(\lim\limits_{\text x \to0}\cfrac{\text x\,tan\,\text x}{1-cos\,\text x} \) = 2