Question

Find the limit as x approaches infinity of `y=arccos((1+x^2)/(1+2x^2))`?

Answer 1

There is a law of limits that deals with composite functions. Essentially, if we have two functions `f` and `g`:

`lim_(x->a) f(g(x)) = f(lim_(x->a) g(x))`

In this case, `f(g(x))` would be `arccos(g(x))`, and `g(x)` would be `(1+x^2)/(1+2x^2)`.

So, to find the limit of the entire thing as `x` approaches infinity, we can find the limit of the inner function as `x` approaches infinity, and then evaluate the outer function with this value:

`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 `lim_(x->infty) (1+x^2)/(1+2x^2)` is equal to `1/2`.

So, we have:

`lim_(x->infty) arccos((1+x^2)/(1+2x^2)) = arccos(1/2)`

The arccosine of `1/2` is equal to `pi/3`:

`lim_(x->infty) arccos((1+x^2)/(1+2x^2)) = pi/3`