Question

How do you use the Squeeze Theorem to find `lim (arctan(x) )/ (x)` as x approaches infinity?

Answer 1

The limit is zero (see the reason using the Squeeze Theorem below).

Explanation:

Since `0 < arctan(x) < pi/2` for all `x>0`, we can say that `0 < arctan(x)/x < (pi/2)/x` for all `x>0`.

But `lim_{x->infty}0=0` and `lim_{x -> infty} (pi/2)/x = 0`.

Hence, by the Squeeze Theorem, `lim_{x->infty} arctan(x)/x = 0`.