# 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#