What's the value of the following limit?

#lim_(x->0)(1 - cos^n(x))/x^2 , n\in N ,n>=1#

1 Answer
Mar 5, 2018

#n/2#

Explanation:

For #n >= 1#,

#(1-cos^n(x))/x^2 = ((1-cosx)(1+cosx+cos^2x + * * * +cos^(n-1)x))/x^2#

# = ((1+cosx)(1-cosx)(1+cosx+cos^2x + * * * +cos^(n-1)x))/((1+cosx)x^2)#

# = ((sin^2x)(1+cosx+cos^2x + * * * +cos^(n-1)x))/(x^2(1+cosx))#

# = (sinx/x)^2(1+cosx+cos^2x + * * * +cos^(n-1)x)/(1+cosx)#

Now evaluate the limit as #xrarr0#, we get

#(1)(n/2) = n/2#