Question #6dd46

2 Answers
sente
Aug 31, 2016

#lim_(x->0)x^2/(1-cos(x)) = 2#

Explanation:

We will make use of some algebra,the well known limit #lim_(x->0)sin(x)/x = 1#

and the following:

  • if#f(x)# is continuous, then #lim_(x->a)f(x) = f(lim_(x->a)x)#
  • if #f(x)# and #g(x)# have finite limits at #a#, then #lim_(x->a)f(x)g(x) = lim_(x->a)f(x)*lim_(x->a)g(x)#

#lim_(x->0)x^2/(1-cos(x))#

#=lim_(x->0)(x^2(1+cos(x)))/((1-cos(x))(1+cos(x)))#

#=lim_(x->0)x^2/(1-cos^2(x))(1+cos(x))#

#=lim_(x->0)x^2/sin^2(x)(1+cos(x))#

#=lim_(x->0)(x/sin(x))^2(1+cos(x))#

#=lim_(x->0)(sin(x)/x)^(-2)(1+cos(x))#

#=lim_(x->0)(sin(x)/x)^(-2) * lim_(x->0)(1+cos(x))#

#=(lim_(x->0)sin(x)/x)^(-2) * lim_(x->0)(1+cos(x))#

#=1^(-2)(1+cos(0))#

#=2#

Ratnaker Mehta
Sep 22, 2016

#2#.

Explanation:

We use the Trigo. Identity # : 1-cos2theta=2sin^2theta#.

Reqd. Lim. #=lim_(xrarr0) x^2/(1-cosx)#

#=lim_(xrarr0)x^2/(2sin^2(x/2)#

#=lim_(xrarr0) (2(x^2/4))/(sin^2(x/2)#

#=2[lim_(xrarr0){(x/2)/(sin(x/2))}^2]#

#=2[lim_(xrarr0)(x/2)/(sin(x/2))]^2#

#=2*(1)^2#

#=2#, as respected sente has derived!.

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