Find the limit as x approaches infinity of #y=arccos((1+x^2)/(1+2x^2))#?
1 Answer
There is a law of limits that deals with composite functions. Essentially, if we have two functions
#lim_(x->a) f(g(x)) = f(lim_(x->a) g(x))#
In this case,
So, to find the limit of the entire thing as
#lim_(x->infty) arccos((1+x^2)/(1+2x^2)) = arccos(lim_(x->infty) (1+x^2)/(1+2x^2))#
It should be easy to see that
So, we have:
#lim_(x->infty) arccos((1+x^2)/(1+2x^2)) = arccos(1/2)#
The arccosine of
#lim_(x->infty) arccos((1+x^2)/(1+2x^2)) = pi/3#