How do I evaluate: #lim_(x->oo) cos^(-1)((1+x^2)/(1+2x^2))#?

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Answer 1 · 2017-10-13T02:35:51

We can play with a little algebra here to make the limit easier to evaluate:

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

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

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

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

As #x->oo#, the first terms in the numerator and denominator go to zero.

Thus:

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

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

#=cos^-1(1/2)#

#=pi/3#