How large must a be for the improper integral to be less than .001?

[Omega_Prime]

Been doing some calculus review to knock the rust off for this coming fall semester and I got stuck...

Homework Statement

From Stewart's book (Early Transcendentals: 6E): (7.8 pg517 #69)

Determine how large the number "a" has to be so that:

$\int$$\stackrel{\infty}{a}$$\frac{1}{x^{2}+1}$dx <.001

Homework Equations

None.

The Attempt at a Solution

Ok, I can easily picture the graph and the area under it. I figured I'd integrate, use "a" for my lower bound and "t" for the upper bound, then by using the potential equation it's just a simple matter of solving for "a" while taking the limit of said equation as t goes to infinity.

I managed to get:

arctan (t) - arctan (a) < 1/1000 (I suck at "latex" but this should technically be the limit of those arctans as t -> infinity < .001)

Here is where I think I'm screwing up... I take the tangent of both sides:

tan [arctan t -arctan a] < tan (1/1000) - and I'm stuck, I know I can't just apply the tangent function independently to both parameters giving me t - a < tan (.001) is there some trig identity I'm not thinking of..?

 

[SammyS]

Take the limit as t → ∞ .

$$\lim_{t\to\infty}\,\arctan(t) = ?$$

 

[Omega_Prime]

Ugh... Sorry to waste your time. The limit is pi/2, all I had to do was take the limit and I was already done lol. For some reason I thought there would be no limit for the arctan and that was bothering me.